MECHANICAL 

INTEGRATORS 

INCLUDING 

THE  VARIOUS   FORMS 

OF 

PLAN  I  M  ETERS. 


Prof.   HENRY   S.   H.   S.", 


Reprinted  from  the  Proceedings  of  the  Institution   of 
Civil  Engineers. 

/  ^47  ^ 


NEW  YOEK 

D.   VAN  NOSTRAND,  PUiiJblSHfiR, 

23  MuBBAY  AND  27  Wabebn  ^''""  ■" 

1886. 
mn   19C8 


PREFACE. 


,  Mechanical  aids  to  mathematical  compu- 
tation have  always  deservedly  been  regarded 
with  interest. 

Aside  from  the  labor-saving  quality  which 
most  of  them  possess,  they  have  a  value  aris- 
ing from  the  fact  that  they  represent  thoughts 
of  more  or  less  complexity  expressed  in  mech- 
anism. 

They  are  of  many  kinds,  and  serve  widely 
different  purposes.  The  reader  will  find  in 
this  essay  descriptions  of  many  that  are  use- 
ful directly  or  indirectly  to  engineers. 


k 


Mechanical  Integrators. 


All  measurements  are  made  in  terms 
of  some  fixed  unit.  The  method  may 
consist  of  a  simple  comparison  of  the 
unit  with  the  quantity  to  be  mensured, 
but  wiien  this  cannot  conveniently  be 
done,  some  indirect  means  must  be  em- 
ployed. Indirect  measurements  may  be 
made  by  measuring  some  physical  effect, 
the  magnitude  of  which  is  known  to  be 
a  function  of  the  quantity  to  be  meas- 
ured ;  as,  for  instance,  when  the  length 
of  a  rod  or  wire  is  estimated  by  its 
weight.  Where,  however,  the  unit  in 
terms  of  which  the  measurement  has 
to  be  made,  is  what  is  known  as  a 
derived  unit,  the  indirect  method  gen- 
erally consists  in  measuring  in  terms 
of    the   simple    units    from    w^hich    the 


6 


former  is  derived,  and  performing, 
with  the  results,  the  necessary  calcula- 
tion. An  example  of  this  latter  method 
is  given  in  obtaining  the  contents  of  an 
area,  by  taking  its  length  and  breadth 
and  multiiDlying  them  together,  instead 
of  adopting  the  tedious  process  of  ascer- 
taining, by  direct  comparison,  bow  many 
times  the  unit  of  area  would  be  con- 
tained within  it. 

Now,  such  calculations,  even  when  of 
so  simple  a  kind  as  mere  multiplication, 
often  become  very  inconvenient,  and  a 
large  number  of  instruments  have  been 
designed  for  performing  them  by  me- 
chanical means.  Such  instruments  may 
be  divided  into  two  classes,  one  in  which 
the  final  result  of  conditions  which  vary 
in  an  arbitrary  manner  is  found,  such  as 
the  contents  of  a  surface  or  the  work  of 
a  motor,  requiring  a  process  of  multi- 
plication or  addition;  the  other  in  w^hich 
the  relation  or  ratio,  at  any  instant,  of 
two  such  quantities  is  given,  such  as 
space  and  time  in  the  case  of  velocity, 
requiring  at  each   instant  a   process  of 


division.  The  object  of  the  present  pa- 
per is  to  deal  with  the  theory,  design 
and  practical  applications  to  engineering 
problems  of  the  former  class  alone.  It  may- 
be briefly  stated  that  very  little  has  yet 
been  piractically  done  in  the  use  of  the 
latter.  Quite  recently,  Professor  A.  W. 
Harlacher,  of  Prague,  has  published  an 
account  of  the  instruments  and  methods 
of  Harlacher,  Henneberg  and  Smreker, 
for  gauging  the  velocity  of  a  river  cur- 
rent, the  principle  of  which  is  the  same 
as  that  independently  adopted  by  the 
author  and  others  in  this  country. 

The  conditions  or  data  above  referred 
to,  from  which  the  required  result  has  to 
be  calculated  by  instrumental  means,  are 
obtained  in  two  ways: 

(1)  By  intermittent  or  separate  obser- 
vations and  measurements. 

(2)  By  the  continuous  motion  of  a 
machine  in  connection  with  self-record- 
ing apparatus. 

The  former  is  the  case  in  measuring 
an  area  of  country,  taking  dimensions  of 
a  river  or  embankment   section,  or  ob- 


8 


taming  the  forces  exerted  at  different 
times  by  a  machine  or  body  in  motion. 
The  latter  is  generally  given  in  the  form 
of  a  graphic  record,  an  important  exam- 
ple of  which  is  the  diagram  of  energy 
or  work  taken  from  a  prime  mover.  In 
both  cases  the  result,  whatever  it  be, 
whether  boundary,  area,  volume,  work, 
etc.,  can  be  found  by  calculation,  but 
only  with  an  approximation  to  the  truth, 
depending  upon  the  extent  of  the  calcu- 
lation. The  reason  of  this  is  that  the 
data  of  calculation,  which  are  taken  di- 
rectly in  the  first  case,  or  selected  from 
the  graphic  record  in  the  second,  only 
represent  actual  conditions  more  or  less 
closely,  according  as  the  number  of  data 
so  taken  is  greater  or  less,  and  the  great- 
er the  number  the  greater  is  the  labor 
of  determining  the  result.  The  instru- 
ments discussed  in  this  jDaper  perform 
such  work  mechanically,  with  the  great 
advantages  of  rapidity  of  operation,  ac- 
curacy of  results,  and  without  requiring 
mental  effort  on  the  part  of  the  manipu- 
lator; and  all  this,  moreover,  to  a  great 


extent  independently  of  the  complexity 
of  the  calculation  required.  All  the  re- 
sults the  measurement  of  which  will  be 
considered,  can  be  measured  graphically. 
If  the  observations  have  been  made  sep- 
arately, tliey  can  be  plotted,  whether  in 
the  form  of  a  diagr  im  of  energy,  or  on 
the  plan  or  elevation  of  an  area  or  sec- 
tion, and  the  boundary  can  be  filled  in 
with  a  tolerably  close  approximation  to 
accuracy.  In  the  other  case  the  graphic 
record  is,  or  may  be,  directly  given.  The 
subject,  as  far  as  the  theory  of  the  cal- 
culation goes,  can  therefore  be  studied 
with  reference  to  such  diagrams  without 
the  necessity  of  considering  in  the  first 
case  how  they  were  obtained,  and  it  will 
be  convenient  to  do  this,  and  afterwards 
to  examine  separately  various  examples 
of  their  application.  Such  diagrams  may 
be  drawn  upon  any  kind  of  surface,  and 
an  instrument  for  dealing  with  measure- 
ments upon  that  of  a  sphere  will  be  here- 
after described.  A  plane  surface  may, 
however,  be  employed  upon  which  to  re- 
present all  cases  of  any  practical  import- 


10 

ance,  and  the  question  thus  arises,  What 
are  the  measurements  of  the  nature  un- 
der consideration  which  are  required,  and 
which  can  be  obtained  from  either  a  reg- 
ular or  an  irregular  plane  of  figure? 

Such  measurements  are  of  three 
kinds : 

(1)  The  length  of  its  perimeter  or 
boundary. 

(2)  The  area  of  its  superficial  contents. 

(3)  Its  relation  to  some  point,  line,  or 
other  figure  on  the  surface,  e.  g.^  its  mo- 
ment of  area  or  moment  of  inertia  about 
a  given  line. 

All  these  three  kinds  of  quantities  can 
be  ascertained  by  successive  operations 
of  addition.  The  first  requires  the  ad- 
dition of  elements  of  length,  the  second 
may  be  obtained  by  adding  up  successive 
elements  in  the  form  of  strips  of  area, 
and  the  third  by  adding  products  ob- 
tained by  multiplying  such  strips  by 
some  quantity,  the  magnitude  of  which 
depends  upon  the  position  of  the  other 
point,  line,  or  figure  in  question.  Tak^ 
ing  the  general  case  of  an  irregular  fig- 


11 


ure,  it  is  evident  that  absolute  accuracy 
can  only  be  obtained  when  this  opera- 
tion becomes  that  of  integration  or  sum- 
ming up  of  an  infinite  series  of  indefin- 
itely small  quantities.  Instruments  for 
performing  this  operation  are  therefore 
called  '•  mechanical  integrators.''  In  all 
such  instruments  the  rolling  action  of 
two  surfaces  in  frictional  contact  is  em- 
ployed, for  this,  as  will  be  hereafter 
seen,  enables  the  conditions  of  motion 
to  be  continuously  varied  in  a  way  which 
could  not  be  effected  by  mere  trains  of 
wheel-work,  such  as  form  the  mechanism 
of  some  kinds  of  calculating  machines. 
This  fact  necessitates  something  more 
than  a  mere  discussion  of  the  mathe- 
matical principles  upon  which  the  calcu- 
lations are  performed,  for  though  the 
action  of  an  integrator  may  be  absolute- 
ly correct  as  far  as  its  theory  of  the  per- 
formance of  the  calculation  is  concerned, 
yet  there  is  always  some  instrumental 
error  depending  upon  the  rolling,  and 
also,  as  will  be  seen,  of  the  slipping  of 
the   two  surfaces    in    frictional   contact. 


12 


This  error  may  be  exceedingly  small.  Init 
it  is  a  matter  of  great  importance  to 
ascertain  its  exact  amount,  and  the  sub- 
ject will  therefore  be  investigated  at 
length,  under  the  heading  "Limits  of 
Accuracy  of  Integrators,"  where  an  ac- 
count will  be  also  given  of  the  experi- 
mental results  of  Professor  Lorber,  of 
Leoben;  Dr.  William  Tinter,  of  Vienna, 
and  Dr.  A.  Amsler,  of  Schaflfhausen.  In 
this  investigation  it  will  be  showu  that 
when  integrators  are  examined  upon  the 
mechanical  principles  of  action,  they  are 
all  found  to  belong  to  one  of  two  clasBes. 

(1)  In  which  the  surfaces  in  question 
slip  over  each  other. 

(2)  In  which  only  pure  rolling  motion 
of  the  surfaces  is  assumed  to  take  place. 

The  significance  of  this  mode  of  clussi- 
fication  is  that  it  not  only  leads  to  a  cl&ir 
understanding  of  the  nature  of  the  re- 
sults to  be  expected  from  any  particular 
instrument,  and  teaches  the  best  method 
of  manipulating  it,  with  regard  to  its  po- 
sition  relatively  to  the  figure  to  be  meas. 
ured,  but  it  also  brings  out  prominently 


13 


tlie  meolianiral  principle  uptui  wliicb  tlie 
inventor  has  relied  sometimes,  as  it 
NVouM  appear,  unoonsciouHly,  for  the  ac- 
ruracv  of  tlie  r«->»tiltH  »\jnMt« d  to  be  ob- 
UiintMl. 

It  may  be  here  remarked  that  the  wime 
priiJcipU',  by  which  an  intej^'rator  is  em- 
j)h)yo(l  to  determine  a  result  from  an  au- 
to^'ni]>!iic  record,  may  Ih)  apphed  directly 
to  «»btain  a  continuous  result  from  the 
machine  or  bo<ly  in  motion,  such,  for  in- 
stance, as  an  ordinary  dynamometer  or 
dynamo-electric  machine,  from  which  the 
autoj^niphic  record  was  obtained.  Thus, 
after  discussing'  the  action  of  intejjrators 
for  dealing' with  diagnims,  it  will  (mly  be 
necessary  to  consider  the  mechanical  de- 
tnils  of  the  instruments  for  direct  appli- 
cation to  a  machine,  and  this  will  be 
done  under  the  head  of  *' Continuous 
Intef^rators." 

Coming  now  to  the  consideration  of 
the  actual  measurement  of  the  three  kinds 
of  (piantities,  it  will  be  found  that  the 
lirst  18  very  simple,  and,  in  fact,  the  only 
reas<^)n  whv  it  is  not  convenient  to  meas- 


14 


ure  it,  by  comparing  it  with  the  unit  of 
length  in  the  ordinary  way,  is  its  contin- 
uous change  of  direction. 

The  only  mechanical  method  of  recti- 
fying a  curve,  as  it  is  called,  that  is.  ob- 
taining its  length  as  a  right  line,  is  by 
rolling  a  wheel  along  it.  This  wheel  is 
connected  with  a  suitable  train  of  wheels 
for  recording  the  total  number  of  revolu- 
tions, and  as  the  rolling  circle  of  the 
first  wheel  is  either  a  unit  in  length  or 
contains  a  known  relation  to  this  unit, 
the  length  of  the  curve  or  boundary  is 
given  at  once  by  the  reading  of  the  grad- 
uated wheels.  The  use  of  such  instru- 
ments is  very  ancient,  and  Beckmann,  in 
liis  "  History  of  Inventions,"  describes 
amongst  various  odometers,  one  men- 
tioned in  the  Tenth  Book  of  Vitruvius. 
Such  an  instrument  is  made  upon  a  small 
scale  for  use  by  a  draftsman,  and  in  one 
form  it  is  sometimes  termed  an  "  opisom- 
eter,"  in  another  form  the  chartometer, 
or  Wealemefna ;  it  is  also  employed  upon 
a  large  scale  as  a  road  or  route  meas- 
urer.    The  same  principle  has  been  em- 


15 


ployed  in  one  of  the  latest  forms  of  ane- 
mometer,  in  which  the  plane  of  the  wheel 
is  always  kept  coincident  with  the  direc- 
tion of  the  wind,  while  its  edge  rolls  in 
contact  with  the  recording  surface,  and 
measures  the  total  travel  of  the  wind.  In 
this  class  falls  the  device  suggested  in 
a  letter  to  Xature  by  Mr.  V.  Ventosa, 
of  Madrid,  for  continuously  obtaining 
the  N.,  E.,  S.  and  VV.  components  of  the 
wind,  a  device  which  was  independently 
arrived  at  by  the  author  in  connection 
with  certain  mathematical  principles  re- 
ferred to  hereafter. 

The  object  of  emj^loying  a  rolling 
wheel  is  merely  to  enable  the  direction  to 
be  changed  so  as  always  to  coincide  with 
the  curve  to  be  measured,  and  the  jDrin- 
ciple  is,  therefore,  that  of  direct  unit 
measurement  or  comparison.  The  fore- 
going instrument,  w^hich  evidently  be- 
longs to  the  second  class,  in  which  only 
rolling  motion,  without  slipping,  is  sup- 
posed to  take  place,  forms,  however,  one 
kind  of  mechanical  integrator. 

The   measurement   of    the   other   two 


16 

kinds  of  quantities  is,  as  in  the  case  of 
the  first,  a  problem  of  addition ;  but  in- 
stead of  being  the  addition  of  an  infinite 
number  of  infinitely  short  lines,  an  in- 
finite number  of  continually  changing 
magnitudes  has  to  be  added.  This  is  the 
same  both  for  instruments  required  in  the 
second  kind  of  measurement,  or  *'Area 
Planimeters,"  and  those  required  iu 
the  third  kind,  or  "  Moment  Planim- 
eters." From  the  fact  that  the  chang- 
ing magnitudes  referred  to  are  sim- 
pler to  deal  with  in  the  case  of  cal- 
culating the  contents  of  areas  than  in 
finding  theii'  moments  or  any  mathemat- 
ical results  of  an  equivalent  kind,  the  dis- 
cussion of  the  theory  of  the  two  above 
kinds  of  instruments  is  not  the  same,  and 
will,  therefore,  be  dealt  with  under  two 
separate  headings. 

Area  Planimeters. 

The  area  of  any  plane  figure,  such  as 
ABDE  (Fig.  1),  can  be  obtained  in  the 
following  manner.  Take  any  straight 
lines,  OX  and  OY,  at  right  angles  to  each 


17 


otber,  and  parallel  to  OY  ;  draw  a  series 
of  straight  lines  equidistant  from  each 
other,  dividing  the  figure  into  a  number 
of  strips,  or  elements  of  area.     A  series 


of  rectangles  may  then  be  found,  as  shown 
at  AB,  the  area  of  which  is  equal  to  that 
of  the  corresponding  strip,  so  that,  by 
adding  the  rectangles  together,  the  area 
of  the  whole  figure  is  obtained,  as  adopt- 
ed in  the  common  method  of  finding  the 
area  of  an  indicator  diagram.  The  greater 


18 


the  number  of  strips  the  more  closely 
will  the  liei^-bt  of  the  two  sides  of  each 
tend  to  become  equal  to  each  other,  and 
to*the^height  of  the  corresponding  rect- 
angle. 

Let  A. '^  =  the  width  of  any  element  of 

area  such  as  AB,  at  a  distance 

X  from  0. 
y= height  of  the  corresponding 

rectangle. 
Then  ?/ A  a; = area  of  the  element  AB, 
and  the  sum  of  all  such  elements  of  area 
is  the  area  of  the  figure  AP>DE.     When 
the  number  of  elements  is  increased  in- 
definitely, this  expression  becomes 

/   2/c/ic=area  of  the  figure  ABDE, 

a  and  h  being  the  extreme  values  of  ic, 
i.e.,  the  limits  of  integration. 

It  is  evident,  therefore,  that  the  area 
will  be  correctly  measured  by  an  instru- 
ment in  which  the  recording  wheel  or 
measuring  roller  always  turns  at  a  rate 
proportional  to  the  ordinate  y  of  the 
curve,  while  the  body  from  which  it  de- 
rives  its  motion  moves  at  a  uniform  rate 
along  the  axis  OX. 


19 


Area  plauimeters  have  been  classified 
accordinjjf  to  apparently  different  modes 
by  which  the  operation  of  integration  is 
performed  ;  but,  since  the  action  of  them 
all  can  be  explained  upon  the  foregoing 
principle  of  adding  elements  of  area,  and, 
in  fact,  by  means  of  the  same  notation,  it 
is  not  surprising  that  such  classifications 
are  anything  but  satisfactory.  In  fact, 
in  one  important  kind  of  planimeters,  it 
becomes  doubtful  to  which  class  they  be- 
long, or,  whether  they  should  not  be 
placed  in  two  or  more  classes.  It  is, 
without  doubt,  very  convenient  to  dis- 
tinguish different  planimeters,  and,  there- 
fore, the  names  which  have  been  given 
them  will  be  used  ;  but  this  will  not  de- 
note any  difference  of  principle,  and  the 
classification  which  will  be  adopted  is 
that  already  explained,  and  depends  on 
mechanical  conditions  of  action.  In  what 
follows,  one  mode  of  viewing  the  mathe- 
matical operation  is  adhered  to  through- 
out, and  it  may  be  stated  that  the  object 
of  the  author  has  been  to  make  clear  the 
principles  of  action  of  integrators,  rather 


20 

than  to  obtain  rigid  and  exhaustive  dem- 
onstrations of  their  theory. 

Planimeters   in   which  Slipping  of    the 
Measuring  Roller  Takes  Place. 

From  a  brief  account  of  the  subject  by- 
Professor  Lorber,  it  appears  that  the  first 
recorded  idea  of  a  planimeter  is  attributed 
to  Hermann,  of  Munich,  who  worked  it 
out  with  Lammle.  This  idea  of  Her- 
mann's, which  was  pubhshed  in  1814, 
seems  to  have  fallen  into  oblivion,  for  in 
1827  Oppikofer,  of  Berne,  constructed  a 
planimeter  upon  similar  principles,  and 
it  was  thenceforth  called  after  his  name. 
On  the  other  hand,  Favora  gives  the 
priority  to  Professor  T.  Gonella,  of  Flor- 
ence, who,  in  1828,  without  any  knowl- 
edge of  what  Hermann  had  done,  invented 
and  described  a  very  similar  instrument. 
The  development  of  the  planimeter  seems 
to  have  grown  out  of  the  instrument  of 
Oppikofer,  who,  in  conjunction  with  a 
Swiss  mechanic,  Ernst,  finished  a  plan- 
imeter which  won  a  prize,  in  Paris,  in 
1836.     Important  improvements  are  due 


21 


22 


to  Wettli,  of  Zurich,  who,  with  Starke,  in 
1849,  took  out  a  patent  in  Austria  for  the 
instruments  now  called  the  Wettli-Starke 
planimeter.  Later  on,  in  England,  other 
inventors  (Sang,  Moseley)  worked  at  the 
subject,  but  all  these  instruments  de- 
pended for  their  action  on  the  same  prin- 
ciple, which  is  as  follows  : 

Let  M  (Fig.  2)  be  the  plan  of  a  disk 
rolling  in  contact  with  a  straight  guide 
PQ,  which  is  parallel  to  OX,  and  at  a 
distg-nce  from  it  equal  to  the  radius  of  the 
disk,  so  that  the  plan  of  the  center  of  the 
latter  always  lies  in  OX.  Let  m  be  a 
roller  upon  the  surface  of  the  disk,  gradu- 
ated and  connected  with  wheel-work  and 
an  index,  so  that  the  distance  turned 
through  over  the  surface  of  the  disk  can 
be  read  in  revolutions  or  parts  of  a  revo- 
lution. The  plan  of  the  point  of  contact 
(B)  of  the  roller  with  the  disk  is  always 
made  to  coincide  with  that  particular 
point  on  the  curve  which  is  in  the  line 
drawn  at  right  angles  to  OX,  through 
the  center  C  of  the  disk.  The  plane  of 
otation  of  (m)  which  may  be  called  the 


23 


measuring  roller,  is  always  perpendicular 
to  the  disk  M,  and  the  plan  of  its  axis,  as 
shown  in  the  figure,  is  always  parallel  to 
OY,  so  that,  in  following  the  curve,  it 
slips  backwards  or  forwards  across  the 
surface  of  the  disk,  in  a  direction  parallel 
to  OY.  Suppose  the  disk  to  roll  along 
PQ  for  a  distance  :_.  .r,  eipial  to  the  width 
of  the  element  AB. 

Then  if  //,  =  distance  of  B  from  OX. 
li  =  radius  of  disk  M. 
?•=: radius  of  measuring  roller 

m. 
;ij = consequent  reading   of 
measuring  roller   for   this 
travel  of  disk. 

Then  ^  a  .'' = linear  distance  turned 
through  by  a  point  on  the 
disk  at  the  distance  y,  from 
the  center. 
2 7r;v/ J  =  linear  distance  turned 
through  by  a  point  on  the 
circumference  of  ?/i  /  but 
since  )n  rolls  on  M  these 
distances  are  equal. 

?/ 
Therefore       2  nrx  ^  =  ~Ax, 

or,  ".  =  .'AA^X2;i^; 


24 


but  K — =^  is  a  constant,  which,  by  taking  . 

27rrll 
r  and  R  in  suitable  ratio  may  be  made 
unity. 

Then  n^—y^Ax, 

that  is,  the  reading  of  the  roller  m  meas- 
ures that  part  of  the  area  of  the  element 
above  OX. 

If  the  iDoint  of  contact  be  made  to  fol- 
low round  the  curve  continuously  in  one 
direction,  then,  when  the  portion  of  AB 
below  OX  is  being  measured,  the  disk  is 
moving  in  the  opposite  direction  along 
PQ,  but,  at  the  same  time,  the  roller  is 
turning  in  the  opposite  way  relatively  to 
the  disk  to  that  which  it  was  doing  be- 
fore, since  the  point   of   contact  is  now 
below  C.     The  final  result  of  these  two 
opposite  motions  is  to  cause  the  roller  to 
turn,  as  at  first,  and  so  add  the  result 
given  for  CA  to  what  was  given  for  CB. 
If  the  motion  of   the  disk   ax  for  the 
width  of  AB  be  now  regarded  as  nega- 
tive, and  —?/2=:  distance  AC 
also  ??2=i'6ading  of  roller  for  this  element 
of  area, 


25 


then  by  similar  reasoning  to  that  already 
used, 

and  71  =  71^  +  ^'2~  (Vi  +.'/o)  A  x  —  y  A  x- 
=  area  of  element 

Tliis  reasoning  holds  for  any  possible 
2)Osition  of  the  roller,  or  of  the  axis  OX, 
which  may  be  altogetheivonfcside  the  fig- 
ure, as  it  practically  is  for  the  integra_ 
tion  of  the  portion  DHE.  Then  it  will 
be  found  that  DKH  is  subtracted,  and 
DKHE  is  added,  so  as  to  give  the  re- 
quired actual  area  DHE. 

Inasmuch  as  this  reasoning  is  inde- 
pendent of  the  actual  value  of  the  width 
of  the  element,  and  as  the  vertical  motion 
of  the  roller  m  has  no  effect  in  theory 
upon  the  distance  rolled  through  bj  it, 
therefore  in  the  limit  when  ^aTbecomes 
infinitely  small,  the  actual  value  of  the 
series  of  infinitely  narrow  strips  which 
compose  the  figure  x4.BDE  is  given  by 
the  final  reading  of  the  roller  when  the 
traverse  of  the  boundary  is  completed. 

The  Wettli-Starke  planimeter  (Fig.  3) 


26 

acts  directly  upon  this  principle,  with 
the  exception  that  it  is  the  disk  that 
is  moved  according  to  the  changes  in  y 
instead  of  the  measuring  roller,  and  the 


following  is   a   description   of   the   best 
form  of  this  instrument: 

On  a  base-plate  P  (Fig.  3),  are  three 
guides  SSS,  along  which  a  frame  carrj^- 
ing  the  vertical  axis  of  the  disk  M  can  be 
moved  to  and  fro.     The  disk,  which  is 


27 


made  of  glass  and  covered  with  paper, 
has  two  motions,  one  rectiHneal  along 
the  guides,  and  one  of  rotation  about  its 
axis.  The  motions  are  imparted  to  it  by 
means  of  an  arm  (L),  which  passes 
through  the  roller-guides  igg)  in  the 
frame  carrying  the  disk,  and  rotates  the 
latter  by  means  of  a  German  silver  wire 
{del )  passing  round  a  cylinder  w  upon  its 
axis,  and  attached  by  the  two  ends  to  the 
extremities  of  the  arm.  The  measuring 
roller  (m)  rests  upon  the  surface  of  the 
disk,  being  carried  in  another  frame  (B), 
which  is  hinged  to  the  base-plate.  The 
action  is  as  follows :  the  base-plate  being 
placed  in  juxtaposition  to  the  figure  to 
be  integrated,  any  line  parallel  to  the 
guides,  z.e.,  to  the  direction  of  rectilinear 
motion  of  the  disk,  may  be  taken  as  the 
axis  OY;  and  line  OX,  drawn  through 
the  edge  of  the  roller,  perpendicular  to 
OY,  may  be  taken  as  the  other  axis. 
Then,  as  the  pointer  ^j>  at  the  extremity 
of  the  arm  is  made  to  pass  round  the 
boundary  of  the  figure,  the  disk  wall  be 
turned  through  a  distance  proportional 


28 


to  the  travel  along  OX,  while  at  any  in- 
stant the  roller  {m)  is  at  the  same  dis- 
tance from  the  center  of  the  disk  as  the 
pointer  is  from  OX. 

If  2/^z=CB=:  mean  height  of  element  A. x'  = 
= width  of  element  AB. 
Then,  by  the  reasoning  already  given? 
the  reading  of  the  roller  which  the 
pointer  passes  over  the  upper  boundary 
of  the  element  AB,  is 

71^  —  y^Ax, 
and  the  final  reading  of  the  roller  is 
N=area  of  the  figure  A  DDE. 
Hansen,  in  1850,  still  further  improved 
this  instrument,  and,  in  conjunction  with 
Ausfeld,  introduced  a  different  method  of 
reading  the  result,  and  of  carrying  the 
frame,  this  instrument  being  known  as 
the  Han  sen- Ausfeld  planimeter.  Various 
other  instruments  of  the  same  kind  were 
shown  in  the  Great  Exhibition  of  1851, 
but  in  all,  the  motion  of  the  arm  carrying 
the  pointer  was  "  linear ; ''  that  is,  the 
motion,  which  must  be  possible  in  every 
direction,  is  obtained  by  compounding 
two  rectilinear  movements,  at  right  angles 


29 


30 


to   each  other.      Such   instruments   are 
therefore  called  "  Imear  planimeters." 

Many  different  forms  of  linear  planim- 
eters have  been  suggested,  but  the  only 
modification  of  the  disk  and  roller  which 
it  will  be  worth  while  to  notice  is  the 
cone  and  roller. 

Let  MM',  Fig.  4,  be  the  cone  cor- 
responding to  the  disk  M,  and  rolling 
on  the  edge  of  its  two  bases  in  a  direction 
parallel  to  OX.  Let  the  roller  ni  always 
be  in  contact  with  a  circle  on  the  cone, 
whose  center  B'  is  at  a  distance  CB' 
from  the  apex  C  of  the  cone,  such  that 

CB'  =  SB  =  ?/=mean  ordinate  of  ele- 
ment SB. 
where  the  element  AB  is  being  at  that 
instant  integrated.  Adopting  the  same 
notation  as  before,  when  the  cone  has 
rolled  over  the  surface  through  a  distance 
Ace,  then,  whatever  be  the  angle  of  its 
apex,  the  distance  rolled  through  by  the 
roller  m  is 


XV 


n,=y^^xx^^-^. 


81 


As  might  have  been  anticipated,  the 
expression  is  the  same  as  was  obtained  in 
the  case  of  the  disk,  the  hitter  being  a 
special  case  of  the  cone  when  the  vertical 
angle  is  180°. 

Thus  the  cone  may  be  employed  in- 
stead of  the  disk,  and  such  an  instrument 
was  invented  by  Mr.  E.  Sang,  who,  in 
1852,  published  a  description  of  it,  ac- 
cording to  which  the  action  was  extremely 
accurate,  but  it  does  not  appear  to  have 
come  into  very  extensive  use. 

No  more  instruments  of  the  kind  will 
be  described,  since  they  have  given  place 
to  those  in  which  the  arm  carrying  the 
pointer  turns  about  a  center  or  pole,  and 
which  are,  therefore,  called  "polar  plan- 
imeters." 

In  the  year  1856,  Professor  Amsler- 
Laflbn  invented  and  brought  before  the 
world  the  now  well-known  polar  planim- 
eter  bearing  his  name,  and,  since  then, 
no  less  than  twelve  thousand  four  hun- 
dred of  these  instruments  have  been 
made  and  stnt  out  from  his  works  at 
Schaffhausen.     According  to  authorities, 


32 


which  Professor  Lorber  quotes,  Professor 
Miller,  of  LeobeD,  invented  independently 
a  planimeter  of  this  kind  in  the  same 
year  (1856),  which,  being  made  by  Starke, 
of  Vienna,  is  known  as  the  Miller-Starke 
planimeter.  Previous  to  this,  in  1854, 
Decher,  of  Augsberg,  as  well  as  Bounia- 
kovsky,  of  St.  Petersburg  (1855),  had 
improved  upon  previously-existing  forms 
of  polar  planimeter,  though  it  is  well  to 
note  that  the  planimeter s  already  men- 
tioned as  sent  to  the  Great  Exhibition  of 
1851  from  various  parts  of  Europe,  as 
Italy,  Switzerland,  France,  and  Prussia, 
were  all  linear,  and  no  mention  is  made 
of  polar  planimeters  in  the  jurors'  re- 
port. 

The  Amsler  planimeter  is  shown  in 
Fig.  5.  It  consists  of  two  bars,  (a)  the 
radius  bar,  and  (b)  the  pole-arm,  jointed 
at  the  point  C.  The  tracing  point  p, 
which  now  coincides  with  the  point  B  of 
the  figure  ABDE,  is  carried  round  the 
curve,  and  the  roller  m,  which  partly  rolls 
and  partly  slips,  gives  the  area  of  the 
figure  ;  and  by  means  of  the  graduated 


33 


m 

V 


34 


dial  h,  and  the  vernier  y,  in  connection 
with  the  roller  m,  the  result  is  given  cor- 
rectly in  four  figures.  The  sleeve  H  can 
be  placed  in  different  positions  along  the 
pole-arm  b,  and  fixed  by  a  screw  s,  so  as 
to  give  readings  in  different  required 
units.  A  weight  at  W  is  placed  upon  the 
bar  to  k'eep  the  needle-point  in  its  place, 
but  in  instruments  by  some  other  makers 
T  is  a  pivot  in  a  much  larger  weight, 
which  rests  on  the  paper.  A  recent  minor 
improvement  has  been  to  fix  a  locking 
spring  to  the  frame,  so  that  the  roller 
can  be  held  when  the  planimeter  is  raised 
for  the  purjDose  of  reading  it. 

The  theory  of  the  polar  planimeter  may 
be  simply  deduced  from  that  of  the  disk 
and  roller  thus : 

Let  Fig.  6  represent  the  same  disposi- 
tion of  the  disk  M  with  regard  to  the 
figure  ABDE,  as  in  Fig.  2.  but  now  let 
the  roller  m  move  round  the  edge  of  the 
disk  instead  of  across  it,  its  distance 
from  OX  being  always  the  same  as  be- 
fore, viz. : 

0^  =  SB=y^. 


35 


36 

The  turning  of  m  ioY  a  given  travel,  A  aj, 
of  the  disk  is  found  thus — draw  Iq  (Fig 
6a)  tangent  to  the  disk  at  m^  so  that 

lq=z  A  a:, 
and  draw  Ik  parallel  to  the  axis  of  rota- 
tion of  m^  then  qk  is  the  distance  tui-ned 
through,  and  Ik  is  that  slipped  through 
by  the  edge  of  the  roller  m^  when  the 
disk  has  rolled  through  a  distance  A  x ; 
therefore,  using  the  same  notation  as  be- 
fore, 

qkz:z^7rrn^ 

and  i-= '-  =  ^m^qlk; 

Iq       A  £C  ^  1      ' 

but  in  Fig.  6   ?^=.^::.sin  OS^. 

But  by  similar  triangles 

leretore 
or  n^=y^/\xx 


Therefore         '=—, 

AX       E' 


27rrK 

This  is  the  same  result  as  previously  ob- 
tained, and  it  has  been  given  in  this  way 
because  there  is  an  important  class  of 
planimeters   to   be  hereafter   described. 


37 


combining  the  polar  planimeter  with  the 
disk  and  roller,  in  which  a  principle  is 
employed  which  is  thus  made  obvious. 
Tliis  principle  is  that  the  turning  of  m  is 
exactly  the  same  as  if  it  were  in  contact 
at  the  point  B,  no  matter  in  what  posi- 


I 


tion  it  may  be  along  the  line  through  B 
parallel  to  OX.  The  turning  of  m  thus 
measures  the  area  of  the  element  as  long 
as  y  does  not  change.  If,  however,  the 
value  of  //  changes  so  that  ')n  changes  its 
distance  from  OX,  the  measuring  roller 
is  likewise  turned  a  certain  additional 
amount  from  this  cause  ;  but  this  does 
not  affect  the  correct  reading  of  the  area 
so  long  as  its  first  and  last  positions 
are  equally  distant  from  OX.     The  rea- 


38 


son  is,  that  then  the  roller  has  turned  as. 
much  in  one  direction  in  moving  away 
from  OX  as  it  has  in  moving  towards  it, 
and  this  is  the  case  for  the  initial  and 
final  positions  of  the  pointer  when  the 
complete  travel  of  the  closed  curve  has 
been  made.  Now,  inasmuch  as  the  ve- 
locity of  the  edge  of  the  disk  is  just  the 
velocity  at  which  the  center  has  been 
shown  to  move  along  OX,  the  disk  may 
be  removed  altogether.  The  roller  is 
then  moved  in  contact  with  the  surface 
of  the  figure  and  with  identically  the 
same  amount  of  turning  as  before,  pro- 
vided that  its  plane  of  rotation  is  turned 
so  as  to  make  the  same  angle  with  OX 
(which  is  now  its  direction  of  transla- 
tion), as  it  did  before  with  the  direction 
of  motion  of  the  edge  of  the  disk  in 
contact  with  it.  This  is  the  case  when  it 
is  turned  through  that  angle,  and  then 
its  axis  of  revolution  coincides  with  the 
radius  S^'. 

In  order  to  keep  the  direction  of  the 
plane  of  rotation  always  at  right  angles, 
it  is  only  necessary  to  have  a  rod  or  bar 


39 


/  Vm^ 


/  /<!' 


40 

S^  capable  of  turning  on  a  pin  at  the 
point  S.  The  pin  at  S  is  attached  to  a 
sleeve,  which  can  freely  move  along  a 
guide-bar,  whose  direction  coincides  with 
the  axis  OX.  By  employing  the  bar  qSq^ 
itself  as  the  axis  of  rotation  on  which  ??i 
turns,  the  simple  planimeter  shown  in 
Pig.  7  is  obtained,  in  which  the  point 
of  contact  of  the  measuring  roller  is 
made  to  pass  around  the  diagram. 
The  turning  of  the  roller  771  correctly 
gives  the  superficial  contents.  The  roller 
can  be  moved  to  any  position  on  the  rod, 
such  as  shown  in  dotted  lines,  Fig.  7, 
without  in  any  way  affecting  its  result- 
ant turning,  and  the  former  point  of  con- 
tact of  the  roller  is  replaced  by  a  pointer, 
which  is  made  to  follow  the  curve  instead 
of  the  roller.  Professor  Burkett  Webb 
has  described  to  the  author  a  planimeter 
of  this  form  used  in  the  United  States, 
known  as  "  Coffin's''  planimeter. 

If,  finally,  the  point  S,  instead  of  mov- 
ing along  the  straight  hne  OX,  which 
may  be  considered  as  a  portion  of  a 
circle  with   its  center  infinitely  distant, 


41 


moves  along  the  arc  SZ  (Fig  7),  or  any 
other  arc,  as,  for  instance,  that  with  ra- 
dius TC,  the  instrument  becomes  the  or- 
dinary Arasler  planimeter  (shown  in  dot- 
ted Unes).  This  explanation,  so  far,  is 
based  upon  that  given  by  Sir  Frederick 
Bramwell,  who  has  further  shown  that 
the  change  from  the  motion  in  a  straight 
path  to  that  in  the  arc  of  a  circle  has  no 
effect  upon  the  ultimate  reading  when  the 
complete  travel  around  the  closed  curve 
has  been  made,  and  the  arm  SB  has  re- 
turned to  initial  position.  The  following 
demonstration  of  the  truth  of  this  ap- 
pears, however,  to  have  an  advantage,  in 
that  it  follows  throughout  the  operation 
of  integration,  especially  as  recent  plan- 
imeters  are  more  complicated  than  that 
of  Amsler. 

1st.  Let  point  S  (Fig.  8),  at  the  ex- 
tremity of  the  radius  rod,  move  along 
the  broken  line  STUVWXYZ,  and  from 
these  points  draw  arcs  with  a  common 
radius  =:K,  cutting  the  curve  in  points 
t^t.^,  ii^u^,  t\u„,  &c.,  s  and  z  being  tangent 
points  at  the  end  of  the  curve  from  the 


42 


points  S  and  Z.  The  proof  has  ah'eady 
been  given  that  the  integration  of  the 
complete  portion  stj,^,  taken  separately,  is 
given  by  the  reading  of  the  measuring 
roller ;  so  also  are  given  the  areas  of  the 
various  other  portions,  t^u^y^t^,  &c.  If 
the  separate  portions  were  integrated 
consecutively,  any  arc,  such  as  t^t^,  would 
be  traversed  in  both  directions  by  the 
measuring  roller,  because  it  would  move 
one  way  around  in  traversing  one  figure 
and  the  apposite  way  in  going  around  the 
adjacent  one,  and  the  reading  due  to  the 
arc  would  be  eliminated. 

Thus  the  whole  curve  may  be  inte- 
grated correctly  at  once  without  going 
round  each  separate  portion  formed  as 
above,  even  if  the  point  S  at  the  end  of 
AB  moves  upon  a  broken  line  instead  of 
along  OX.  Next,  substitute  a  continu- 
ous curve  TUVXYZ  (Fig.  8a)  for  the 
broken  line.  This  curve  may  be  sup- 
posed to  consist  of  an  infinite  number  of 
straight  portions.  The  infinitely  small 
portions  contained  between  an  arc,  as 
n^u^,  and  another  very  close  to  it,  drawn 


43 


from  the   beginning  and    end   of    these 
straight  portions,  may,  just  as  in  the  case 


of  the  broken  Hne,  be  supposed  to  be  in- 
tegrated  separately  and  with  a  correct 
result,  which  is  independent  even  of  a 
possible  crossing  of  the  arcs,  as  v^v^  and 


44 

t('^u^.  In  the  same  way  as  before,  it  may 
be  seen  that  the  arcs  u^u.^,  etc.,  need  not 
be  traversed  and,  so  long  as  the  point  S 


returns  to  its  initial  position,  the  area  of 
the  figure  is  given  by  the  simple  traverse 
of  its  boundary,  whatever  be  the  curve 
on  which  the   point  S  moves,  which,  in 


45 


the  case  of  the  Amsler  planimeter,  is  a 
circular  arc. 

Various  writers  have  explained  the  ac- 
tion of  the  simple  polar  planimeter  in 
ways  more  or  less  different.  One  of 
these  ways,  recently  given  by  Mr.  F. 
Brooks,  of  Lowell,  U.  S.,  may  be  alluded 
to.  He  shows  that  the  area  may  be 
treated  as  the  difference  between  the  area 
swept  out  by  the  line  Tp  (Fig.  9)  and  the 
sector  of  the  zero  circle,  or  circle  upon 
whose  circumference  the  pointer  {p)  be- 
ing moved,  the  measuring  roller  is  not  in 
consequence  turned.  This  is  true,  both 
for  the  outside  or  inside,  if  proper  signs 
be  taken.  Let  the  element  of  area  2^9,  ^^ 
passed  over,  the  curve  at  />  being  for  a 
small  distance  considered  concentric  with 
the  zero  circle,  this  small  area  subtending 
an  angle  w  at  the  center;  then,  if  values 
be  taken  as  shown  upon  Fig.  9,  in  which 

CT  — radius- bar  =  (7  ; 

Cp  =:one  portion  of  pole- arm  =r^  ; 

Gm  =  i\\e    other  portion    of    pole-arm 

kxeo.  2)q  =  \io  {a^  +  lr  -\-1ab  cos  6)  —^ro 
{cr  +  b'-{-2cb)=2ob  {a  cos  O-c)^ 


46 


and,  by  a  geometrical  construction,  the 
travel  of  the  measuring  roller  is  easily 
shown  to  be  equal  to  the  same  expres- 
sion. Mr.  Brooks  also  explains  why  the 
area  of  the  zero  circle  must  be  added  to 
the  reading  if  the  figure  to  be  integrated 
contains  the  center  T.  The  following  ap- 
pears, however,  to  be  a  still  simpler  ex- 
planation. Keferring  to  Fig.  7,  it  is  evi- 
dent that  going  around  the  outside  of 
the  zero  circle  corresponds  to  a  move- 
ment taken  continuously  above  the  zero 
line  OX  when  only  the  portion  above  OX 
is  measured.  In  this  latter  case  the 
curve  could  never  be  completely  trav- 
ersed as  long  as  the  pointer  nioves  in 
one  direction.  Suppose,  however,  that 
the  ends  of  OX  are  bent  round  and 
brought  withiu  the  figure,  then  the  mo- 
tion in  one  direction  will  enable  the  com- 
plete circuit  to  be  made  ;  but  only  the 
portion  outside  the  line,  i.  e.,  correspond- 
ing to  that  originally  above  OX,  will  be 
measured  by  the  roller,  and  that  within 
must  consequently  be  added  to  the  re- 
corded result.     This  quantity  is  evident- 


48 


ly  the  area  of  the  zero  circle  in  the  case 
of  the  Arasler  planimeter,  which  must, 
therefore,  always  be  added  to  the  result 
when  the  center  of  the  radius- arm  is 
within  the  diagram  to  be  measured. 

As  the  Amsler  planimeter  alone,  so 
far  as  the  author  is  aware,  has  been 
modified  to  measure  the  area  of  any  non- 
developable  surface,  this  modification 
may  be  here  noticed.  The  only  surface 
of  the  kind  to  which  it  has  been  adapted 
is  a  spherical  one.  Fig.  10  shows  the 
instrument,  and  from  that  figure  it  will 
be  seen  that  the  chief  alteration  is  the 
placing  of  two  joints  j  j\  one  upon  the 
radius-bar  («),  and  the  other  upon  the 
pole-arm  h,  so  as  to  allow  the  employ- 
ment of  the  integrator  for  surfaces  of  va- 
rying curvature.  The  joints  are  equi- 
distant from  the  end  of  each  bar,  and 
exactly  opposite  to  each  other— the  ra- 
dius-bar and  pole -arm  being  now  of  equal 
length,  and  a  pin  /  is  placed  on  («), 
which  fits  into  a  corresponding  recess  in 
[h),  so  that  when  the  two  arms  are  closed, 
they  can  be  together  bent  at  the  joints 


49 


to  the  required  amount,   and,  the  joints 
being  purposely  made  stiff,  they  will  re- 


i 


main  at  the  proper  angle  when  the  in- 
strument is  used.  The  joint  (J)  acts  so 
that  the  tracing  point  (p)  is  always  in  the 
place    of    the   axis    of   rotation    of    the 


50 


measuring  roller.  The  theory  of  the  ac- 
tion of  this  instrument  has  been  fully 
exj^lained  by  Professor  Amsler,  in  an  ar- 
ticle in  which  the  theory  of  the  relations 
between  measurement  upon  a  spherical 
surface  and  upon  a  plane  surface  is  dis- 
cussed. 

The  various  applications  of  the  simple 
planimeter  for  finding  areas  are  well 
known,  and  need  not  be  explained  ;  but 
there  are  some  slight  modifications  of  the 
instrument  for  special  purposes,  and  one 
of  these  recently  applied  by  Professor 
Amsler  to  his  planimeter  is  worth  notic- 
ing. This  is  illustrated  at  Fig.  11,  and 
is  a  device  for  readmg  at  once  the  mean 
pressure  given  from  an  indicator  dia- 
gram. Two  points,  U  and  V  are  seen, 
one  (IT)  being  upon  the  upper  side  of  the 
bar  A,  which  slides  in  the  tube  H,  and 
one  (V)  upon  the  tube  H  itself.  These 
points  can  be  adjusted  to  the  length  of 
the  diagram  by  inverting  the  instrument 
in  the  way  shown  in  the  figure,  and  the 
sliding-bar  is  then  clamped  by  the  screw 
S.     This   setting  causes  the  reading  of 


i 


51 


52 


the  instrument,  when  the  diagram  is 
traversed  by  the  pointer  in  the  usual  way, 
to  give  at  once  the  mean  height  of  the 
diagram  in  fortieths  of  an  inch.  The 
simple  relation  is  as  follows : 

Beading  of  measuring-rollers 

=  Mean   height     of     dia- 
gram in  inches. 
Mean   pressure  =  Mean    height  X  vertical 
scale  of  diagram. 

As  an  instance  of  the  great  saving  of 
labor  by  the  use  of  the  Amsler  planimeter, 
the  author  happens  to  know  a  civil  en- 
gineer's office,  where  a  large  amount  of 
earthwork  quantities  had  to  be  taken 
out,  the  calculations  proceeded  slowly 
and  with  many  repetitions,  until  one  of 
the  draftsmen  procured  a  planimeter, 
and  then  the  other,  with  the  result  of  a 
great  expedition  of  the  work,  and  the 
almost  complete  absence  of  errors — and 
even  then  only  in  decimal  places— where 
previously  the  divergence  had  been  as 
much  as  by  units. 

Although  the  connection  between  the 


53 


disk  and  roller  or  linear  planimeter  and 
the  polar  planimeter  has  been  shown,  it 
is  possible  to  regard  them  as  acting  upon 
different  principles.  The  former  may  be 
considered  as  measuring  the  variation  in 
the  ordinate  (y)  by  a  change  of  effective 
radius  of  the  circle  on  which  the  measur- 
ing-roller works,  the  latter  measuring  the 
same  thing  by  a  coiTesponding  change  in 
the  sine  of  the  angle  which  its  plane  of 
rotation  makes,  with  its  direction  of  mo- 
tion over  the  surface  on  which  it  rolls. 
They  have,  in  fact,  been  classified  in  this 
way  as  radius  machines,  and  sine  or 
cosine  machines,  for  the  slipping,  al- 
though occurring  in  both,  appears  in  the 
ordinary  way  of  viewing  the  subject  to 
affect  the  result  in  different  ways.  In  the 
former,  slipping  is  sui)posed  to  be  entirely 
due  to  the  variation  in  the  value  of  (y), 
and  only  takes  place  when  the  ordinate 
changes  in  value :  in  the  latter,  the 
change  is  supposed  to  be  effected  by 
turning  the  pole-arm  about  its  center, 
without  any  slipping  at  all.  This  dis- 
tinction is,  however,  quite  an  imaginary 


54 


one,  for  it  will  be  seen  that  if  the  curve 
be  obliquely  inclined  in  either  case  to  the 
axis  OX,  the  action  of  the  measuring 
roller  is  precisely  the  same.  Recently,  a 
large  number  of  what  are  called  "  Pre- 
cision Polar  Planimeters,"  have  been  de- 
signed and  constructed,  which  combine 
in  an  obvious  manner  the  above  two  prin- 
ciples of  action,  the  disk  giving  motion 
to  the  roller,  while  the  pole-arm  carries 
it  across  the  disk  in  an  oblique  direction. 
Thus,  the  advantages  of  a  uniform  and 
invariable  surface  of  contact  for  the  roller, 
and  the  convenience  of  the  polar  plan- 
imeter  are  combined,  with  the  still  more 
important  advantage  of  a  large  relative 
turning  of  the  measuring  roller.  Before 
describing  a  few  different  forms  of  the 
best  of  these  instruments,  the  general 
theory  upon  which  they  work  will  be 
given ;  it  will  then  not  be  difficult  to  un- 
derstand the  action  of  the  several  instru- 
ments without  repeating  the  exi)]anation 
in  each  case.  It  will  be  found  that  both 
the  linear  and  polar  planimeter  are  only 
special  cases  of  application  of  the  general 


55 


princijDle  upon  which  the  correctness 
of  action  of  j^recision  planimeters  de- 
pends. 


c 


I   2            |0 

&\ 

al 

1°^ 

It  will  be  well  to  approach  the  matter 
from  the  same  point  of  view  as  in  explain- 
ing the  lineal-  planimeter.  Let  the  disk 
M,  Fig.  12,  rotate  about  an  axis  C  as  it 


56 

rolls  along  the  PQ,  line  parallel  to  OX, 
the  pivot  on  the  axle  at  C  being  attached 
to  a  frame  which  also  carries  another 
pivot  S.  This  latter  pivot  always  lies 
upon  OX,  and  about  it  rotates  a  pole-arm 
b,  carrying  a  pointer  j)  at  one  end,  and 
the  measuring-roller  m  at  the  other  end. 
The  plane  of  rotation  of  the  measuring- 
roller  coincides  with  the  .direction  of  the 
pole-arm,  and  is  carried  over  the  disk  in 
contact  with  it,  along  th^  arc  rmn' .  Then 
from  what  was  proved,  p.  402,  the  motion 
of  the  roller  in  is  exactly  the  same  as  if 
it  were  moved,  so  that  its  axis  {ihvays  co- 
incides with  Qj,  the  perpendicular  upon 
the  pole-arm  from  the  center  of  the  disk 
— provided  only  that  its  axis  is  always 
parallel  to  this  line.  Thus,  adopting  the 
previous  notation,  and  taking 

8^:*= length  of  upper  portion  of  pole- 
arm  =R. 
Then  when  the  disk  rolled  through  a 
distance  ax, 
n^^reading  of  roller  m 
2/j  =  ordinate  SB. 


57 


turning  of  roller        _Jl7trn^ 
distance  turned  by  edge"    i\x 


therefore         ^  —  ^i 

AX      K 


or  7? 


wliich  is  the  same  result  as  in  the  case  of 
both  the  linear  and  polar  planimeters.  In 
practice,  the  portion  of  the  pole-arm 
which  carries  the  poii>ter  is  usually  per- 
pendicular to  the  other  portion,  as  shown 
by  the  dotted  lines,  Fig.  12.  In  this 
case,  the  direction  of  motion  of  the  disk 
and  frame  carrying  the  center  of  the  pole- 
arm  8  must  be  taken  parallel  to  the  guide 
P'Q',  that  is,  perpendicular  to  the  former 
direction.  It  has  already  been  shown  in 
the  case  of  the  Amsler  planimeter,  that  it 
does  not  matter  in  what  path  the  center 
S  of  the  pole-arm  is  carried,  so  long  as 
the  foregoing  conditions  are  observed, 
and  thus  there  are  several  forms  of  pre- 


58 


cision  polar  planimeters  in  which  the 
point  S  is  carried  in  the  arc  of  a  circle 
instead  of  along  a  straight  line.  It  may 
now  be  made  clear,  from  Fig.  13,  that 
the  first  two  kinds  of  planimeters  are 
special  cases  of  the  last. 

(A)  Fig.  13.  Let  R  be  the  radius  of 
the  disk,  Rj  the  radius  about  which  the 
roller  in  is  carried.  Then  the  area  of 
the  diagram  as  already  explained  can  be 
measured  by  either  pole-arm  S/)  or  S^:)'. 

(B)  Fig.  13.  Let  the  radius  R,  of  the 
pole-arm  become  infinitely  great,  while  R 
remains  finite;  thence  on  moves  across 
the  disk  M  in  a  straight  line  usually,  but 
not  necessarily,  through  the  center,  and 
the  linear  planimeter  is  the  result. 

(C)  Fig.  13.  Let  the  radius  of  the 
disk  become  infinitely  great,  and  any 
motion  of  such  an  imaginary  disk  under 
these  conditions  would  make  the  result 
equivalent  to  moving  the  roller  over  the 
surface  of  the  paper.  Therefore  the  disk 
may  be  removed,  and  the  elementary 
form  of  polar  planimeter  is  obtained,  the 
roller  being    placed   in   either    position 


59 

Fis.  13 


^(C) 


60 

as  shown  at  m  or  m,  without  affectmg 
the  result. 


^Ar 


7-. 


I       / 


'   '/ 
\  1/       / 


^"^-J 


The  following  is  a  simple  explanation 
of  the  action  of  the  precision  polar  plan- 
imeter 


61 


Let  M  (Fig.  14)  be  the  disk,  which  can 
be  turned  by  any  suitable  means  through 
a  distance  corresponding  to  the  hnear 
travel  of  its  center  about  C. 

Let  r„  =  radius  of  zero  circle  (E'ES')- 

7'= radius  of  any  circle  FF'. 

a=:/ turned  through  by  pole- 
arm,  when  the  pointer  moves 
from  the  zero  circle  to  the 
circle  FF'. 

9''= /turned  through  by  radius 
arm,  CS,  when  an  element 
EE'  F'F  is  being  described. 

a  =  radius  arm  =  CS'. 

i  =  pole- arm  =  FS'. 

d=SS\ 

Then  from  the  figure — 

and  r;  =  a'  +  ^r 

r'  =  a'  +  h'-2ab  cos  (90  + a) 
=  a'  -{-b"^  -{-2((b  sin  a. 
Therefore  7-^  =  9\'^  +  2ab  ^n  a, 

or  sm  a=—-—-^ 

2ab 

Now  the  turning  of  the  plate  is  pro- 
portional to  c'',  and  may,  for  the  arc  FF', 
be  taken  as  equal  to  r^(J'c. 

y=SK  =  SS'  sin  ZSS'K=d  nin  a. 


62 


where  c  and  d  are  constant  quantities ; 
also  if  ft  equal  the  radius  of  the  roller. 
,      y  _  linear  distance  by  ed^e  of  roller 
K     distance  traveled  by  edge  of  disk 

■'■"=^^' 

7\(,''C  lab 

^''  —  ^\\/     ^^^^    \ 


7*  ccl 
^ut?r— 4^ — ris   a   constant   quantity,  and 

anay  be  made  equal  to  unity. 

r'  — r  ' 
Therefore  oi^^ — -^-^^^area    of   ele- 
ment EETF'. 

Thus  n,  is  a  measure  of  an  element  of 
area,  and  as  the  motion  of  m  due  to  the 
turning  of  the  pole- arm  in  moving  to  a 
larger  or  smaller  cu'cle  does  not  affect 
the  reading  when  the  pointer  at  F  has 
passed  round  a  closed  curve,  the  final 
reading  of  the  roller  gives  the  area  of 
any  figure. 


63 

The 

actual   construction 

of 

the 

pre- 

cision 

polar 

l^lanimeter  appears 

to 

have 

\M 

ITuy..  lo 

V 

^ 

S 

a 

c 

I-    ^ 

^— t-^Li 

& 

Fi-.  IG 


been  first  carried  out  by  Mr.  Hohmann, 
Bauamtmann  of  Bamberg,  in  1882,  in 
conjunction  with  the  well-known  mechan- 


64 


ician  Mr.  Coradi,  of  Zurich.  A  plan  of 
the  first  instrument  is  given  in  Fig.  16  ; 
but  it  will  be  more  easily  understood  by 
reference  to  the  diagram,  Fig.  15,  which 
shows  a  frame  (a)  pivoted  at  one  end  (c) 
to  a  weight  (G),  about  which  it  turns. 
This  frame  carries  a  small  disk  (ic),  which 
rolls. in  contact  with  the  surface  of  the 
diagram,  and  gives  motion  to  the  disk 
M.  The  roller  771  is  moved  across  the 
disk  in  the  horizontal  direction  by  a 
pole-arm  centered  on  S  as  axis.  Re- 
ferring to  Fig.  16,  which  is  lettered  in 
a  similar  way  to  Fig.  15,  it  will  be  seen 
how  the  pole-arm,  in  turning  about  the 
center  S,  effects  this  motion.  A  plan  and 
elevation  of  the  frame  F,  which  carries 
the  roller  m,  is  shown  on  a  larger  scale, 
and  this  frame  is  moved  backwards  or 
forwards  through  a  slot  in  the  support- 
ing frame.  The  roller  m  has  the  ar- 
rangement of  the  screw  and  worm  for 
obtaining  the  readings  of  the  dial  //,  as 
in  the  Amsler  planimeter,  and  also  the 
vernier  in  conjunction  with  the  measur- 
ing  roller.     Two   rollers,  j  j\    serve    to 


65 

Fi-.  IT 


Uii^J — p[L^ 


M     '"(©^ 


i 


E 


%^^^€PI 


iM 


Q  SECT! 


ONAL  ELEVATION 


FiiX.  lO 


i)>~tt@ 


PLAN 

fw:TH  Disk  removed) 


66 


balance  the  instrument.  The  details  of 
the  arrangement  by  which  the  length  of 
the  pole-arm  b  is  adjusted  are  also  shown 
on  a  larger  scale. 

In  this  instrument,  the  fact  that  the 
disk  is  inclined  at  an  angle  makes  no 
diffei'ence  in  the  theory  of  its  action,  and 
as  the  roller  W  obviously  drives  the  disk 
so  that  the  angular  motion  corresponds 
with  the  angular  motion  of  the  radius  bar 
a,  the  explanation  already'  given  makes 
its  mode  of  operation  clear.  The  case  is 
rather  simplified  by  the  fact  that  the 
roller  m  is  moved  radially  across  the 
disk. 

An  instrument  of  similar  kind  has  been 
designed  and  recently  described  by  Pro- 
fessor Amsler-Laffou.  This  is  shown  in 
Figs.  17,  18  and  19,  where  it  will  be  seen 
that  this  disk  M,  which  is  now  horizon- 
.tal,  is  turned  by  means  of  bevel-wheels 
Sj  z^,  the  back  of  one  of  which  forms  a 
portion  of  a  frustum  of  a  cone  rolling 
about  the  center  C  of  the  radius-arm  a  a. 
The  center  is  itself  a  sphere,  which  al- 
lows any  side  motion  of  the  instrument 


67 


due  to  the  inequality  of  the  surface  to 
take  place  without  affecting  the  accuracy 
of  the  result.  The  necessary  pressure  cf 
the  roller  upon  the  disk  is  obtained  by 
allowing  the  weight  of  the  portion  of  the 
frame  b  which  carries  the  roller  in  to  rest 
upon  the  disk  by  being  pivoted  by  the 
centers  KK  (Fig.  18).  A  peculiar  fea- 
ture of  the  instrument  is  that  the  pole- 
arm  frame  can  be  centered  either  within 
or  without  the  frame.  If  placed  in  the 
former  position,  the  reading  is  twice  as 
great  as  in  the  latter,  the  positions  of 
the  centers  being  purposely  adjusted  to 
effect  this.  The  frame  can  be  taken  off 
one  center,  s'  (Fig.  17),  by  unscrewing  a 
set  screw  at  x,  and  at  once  placed  upon 
the  other.  The  weight  v^  can  be  adjusted 
in  any  position  by  means  of  the  nut  and 
screw  t  (Fig.  17),  and  so  the  pressure  of 
the  pointer  p  upon  the  surface  of  thQ 
diagram  may  be  regulated. 

In  both  the  above  instruments  the 
disks  derive  their  motion  from  a  roller 
in  contact  with  the  surface  of  the  dia- 
gram, but  in  the  next  two  instruments  to 


68 


be  described,  Messrs.  Holimann  and  Co- 
radi  have  caused  the  disk  to  be  turned  in 
a  manner  which  prevents  any  such  error 
as  from  the  possible  sHpping  of  the  above 
roller.  The  first  instrument  of  the  kind 
is  shown  (Figs.  20,  21)  in  plan  and  ele- 
vation. The  disk  M  is  carried  by  a  frame 
(aa)  as  before,  but  the  frame  now  swings 
about  a  circular  stand,  the  edge  of  which 
is  toothed,  so  that  the  pinion  (^),  which 
is  upon  the  axis  of  the  disk,  is  turned, 
and  therefore  the  disk  itself,  with  the 
same  angular  velocity.  The  weight  of 
the  frame  and  disk  is,  to  a  great  extent, 
taken  off  by  means  of  the  light  rod  [l), 
which  swings  about  a  central  pillar  P. 
A  side  view  of  the  pole-arm  is  shown, 
and  the  mode  of  adjusting  it  and  sup- 
porting the  portion  which  carries  the 
roller  (m),  so  that  by  means  of  centers 
KK  the  weight  of  that  portion  of  the 
frame  is  allowed  to  rest  upon  the  disk. 

It  is  evident  that  this  instrument 
works  upon  identically  the  same  princi- 
ples as  the  foregoing  ones.  This  "Freely 
swinging  "  precision    planimeter  was  fol- 


69 


Fi^f^.  i20  aiid  21 


SECTIONAL  ElE.'ATiON 


70 


lowed  by  another,  called  the  ''  Plate  " 
planimeter  (Fig.  22),  which  is  of  still 
simpler  construction.  In  this  form  ad- 
vantage is  taken  of  the  fact  that  the 
measuring  roller  need  not  have  its  path 
through  the  center  of  the  disk,  and  a  sup- 
port {v)  is  obtained  above  the  disk,  so 
that  its  pivot  [q)  can  work  between  cen- 
ters, the  weight  of  the  frame  being  sup- 
ported by  rollers  {j).  The  portion  of 
the  pole-arm  which  carries  the  roller  {711) 
is  (as  in  the  last  case)  pivoted  between 
the  centers  KK.  The  dial  for  higher 
readings  is  as  in  the  case  of  the  previous 
instruments  denoted  by  A. 

The  last  and  most  recent  modification 
is  the  "  Rolling  planimeter,"  of  Coradi. 
This  approaches  nearest  to  the  diagram 
(Fig.  12),  which  completely  explains  its 
action.  Here  the  center  of  the  radius- 
arm  is  removed  to  an  infinite  distance, 
and  the  center  of  the  disk  and  that  of  the 
pole  arm  are  carried  along  straight  lines 
parallel  to  the  axis  OY  in  that  figure. 
The  way  in  which  this  is  efi'ected  is  seen 
from  Figs.   23  and  24,  which  show  Co- 


71 


radi's  rolling-  planimeter  in  plan  and  ele- 
vation. Two  rollers  {ec/)  are  in  contact 
with  the  surface  of  the  diap^ram,  i  n 


ITiiZ. 


SECTIONAL  ELEVATION 


their  axis  is  a  bevel  wheel  (z^)  (Fig.  24), 
which  gears  with  another  bevel-wheel  (2,), 
which  is  upon  the  axis  of  the  disk.  Thus 
the  wheels  2,  2,  are  turned  as  the  frame 
is   rolled   along,    and,  consequently,  the 


73 


disk  itself.  The  axis  of  the  rollers  cc 
works  upon  the  centers  ee^  which  are  set- 
screws  in  the  frame  (aa).  The  disk  M 
is  also  carried  between  centers  {qq)-,  as  in 
the  instrument  last  described,  and,  also, 
as  in  that  case,  the  path  of  the  roller  does 
not  pass  throuf^h  the  center  of  the  disk. 
This  instrument,  which  has  many  advan- 
taj^es,  and,  notwithstanding:^  that  it  rolls 
on  the  diagram  surface,  <]fives  results  of 
great  accuracy,  has  been  examined  with 
great  care  by  Professor  Lorlier,  who  has 
given  a  lengthy  description  of  it  and  a 
full  account  of  its  theory. 

The  last  planimeter  of  this  kind  to  be 
examined  is  one  by  Professor  xVmsler. 
This  instrument,  shown  in  Figs.  25,  26, 
27,  (lifters  from  the  last  in  that  the  tooth- 
wheel  z,^  works  in  gear  with  a  rack  2,  2,, 
which  is  cut  upon  a  fixed  frame  DD. 
Thus,  although  it  is  supported  by  the 
rollers  cr,  there  is  no  possibility  of  slip- 
ping as  far  as  the  turning  of  the  disk  is 
concerned.  The  rollers  run  in  a  groove 
cut  in  the  frame  DD,  and  the  action  of 
the  instrument  is  easy  and  smooth.    The 


74 


Fio-.  24 


ELEVATION 


Kiss.  ;ao,a6  and  '^T 


75 


theory  of  its  action  is  identical  with  that 
of   the   foregoinj^   one,   as  explained  by- 


means  of  the  diagram  (Fig.  12).  The 
various  parts  are  lettered  in  the  figures 
to  correspond  with  the  explanations  of 
thatjnstniment  previously  given. 


76 


In  the  instruments  hitherto  described 
the  surfaces  of  revohition  are  Hmited  to 
the  disk  and  cone,  but  various  other  sur- 
faces may  be  made  to  replace  these.  The 
only  one  that  has  been  so  employed  is 
that  of  the  sphere;  and  in  the  present 
class  of  instruments,  in  which  slipping 
takes  place,  the  following  property  of  the 
sphere  is  made  use  of :  Let  a  sphere  M 
(Fig.  28),  which  replaces  the  disk  (Fig. 
2),  roll  along  the  axis  OX.  Then  sup- 
pose the  roller  m  can,  by  suitable  means, 
be  moved  round  the  surface  so  that  its 
plane  of  rotation  shall  always  contain  the 
center  of  the  sphere  and  be  perpendicular 
to  the  arm  CB,  which  corresponds  to  the 
pole-arm  of  the  former  instruments  ;  it  is 
evident  that  if  the  perpendicular  be 
drawn  from  q,  the  jjoint  of  contact  of  m 
with  the  sphere,  to  CZ  the  axis  of  rota- 
tion of  the  sphere,  meeting  it  in  the  point 
u,  the  line  qu  is  the  radius  of  the  rolling 
circle  of  contact  of  the  measuring  roller. 
_,,  »  motion  of  measuring  roller  _ 
motion  of  sphere  along  OX  ~ 
qu      qu 


77 


But  from  the  figure  ^^3^=^  — sin  a. 

Therefore,  adopting  the  same  notation  as 
hitherto  used, 

motion  of  m _27rr)i ^  __y 
motion  of   .\l        A  x       K 

which  proves  that  the  area  of  the  curve 
may  be  measured  by  any  device,  on  the 
principle  of  Fig.  28.  It  may  be  shown 
in  the  same  way  as  on  p.  18,  that  the  re- 
sult is  similar  if  the  sphere  rolls  upon  the 
arc  of  a  circle,  about  any  center  as  T, 
instead  of  along  the  straight  line  OX. 
Planimeters  of  this  kind  have  been  con- 
structed by  Mr.  Hohmann  and  Professor 
Amsler.  In  both  cases  only  portions  of 
the  whole  surface  of  a  sphere  have  been 
employed,  and  the  motion  is  given  by 
means  of  an  axis,  instead  of  by  rolling  the 
spherical  surface  upon  the  diagram.  In 
Mr.  Hohmann 's  planimeter,  shown  in 
plan  and  elevation,  Fig.  29,  the  concave 
surface  M  is  used.     Rotation  is  given  to 


SECTIONAL  ELEVATION 


79 


this  by  means  of  an  axis  gr/,  an  enlarged 
portion  of  which  (r)  rolls  npon  a  cii'cular 
metal  path  R.  The  pole-arm  P  turns 
about  a  center  r,  and  so  causes  the  rolling 
circle  of  the  measuring  roller  iit  to  ^'ary 
according  to  the  foregoing  principles. 
This  instrument  has  not  come  into  use. 
Professor  Amsler  has  employed  the  con- 
vex surface  in  an  instrument  somewhat 
similar  to  the  one  described,  except  that 
better  provision  is  made  for  obtaining 
the  required  pressure  between  the  sur- 
face of  the  roller  and  sphere,  and  for 
giving  rotation  from  the  roller-path. 


80 


Planimeters  in  which  only  Pore  Eolling 
Motion  is  Assumed  to  take  Place. 
There  have  been  many  efforts  to  de- 
sign instruments  in  which  no  sHpping 
shall  take  place.  These  efforts  have  re- 
sulted in  the  production  of  various  in- 
struments which,  though  they  differ  in 
external  form  and  mechanical  action,  yet 
rely  upon  the  same  mathematical  prin- 
ciple of  action  as  the  planimeters  al- 
ready dealt  with,  the  particular  form  of 
disk  and  roller,  or  sphere  and  roller,  be- 
ing taken.  Thus,  in  every  case  there  is 
a  measuring  roller,  or  its  equivalent,  the 
rate  of  motion  of  which  has  to  l^e  varied 
by  some  means  or  other.  It  is  in  the 
method  by  which  this  is  done  that  this 
class  of  planimeters  differs  from  the 
other.  Instead  of  obtaining  the  varia- 
tion of  the  measuring  roller  in  bringing 
it  into  contact  with  circles  of  different 
linear  velocity  by  sliding  it  over  the  sur- 
face of  the  disk  or  sphere,  one  or  other 


81 


of  the  two  following  principles  are  em- 
ployed. A  device  equivalent  either  to 
(1)  bringing  in  succession  a  series  of 
measuring  rollers  into  contact  with  the 
different  imaginary  circles ;  or  (2)  bring- 


ing circles  of  different  linear  velocity  in- 
to contact  with  a  single  fixed  measuring 
roller. 

The  disk-globe  and  cylinder-integrator 
of  Professor  James  Thomson  belongs  to 
the  former  class.  In  this  a  sphere  G 
(Fig.  30)  rolls  over  the  surface  of  the  disk 


82 


M,  but  is  also  in  contact  with  a  cylinder 
mm'.  The  motion  of  G  in  direction  OY  is 
that  in  which  the  roller  would  shp  in  the 
ordinary  disk  and  roller,  and  does  not 


31 


affect  the  motion  of  rotation  of  mm' .  On 
the  other  hand,  the  motion  in  direction 
OX,  which  is  due  to  the  turning  of  the 
disk,  is  entirely  imparted  to  nuti.  Thus, 
as  G  rolls  along  m7n\  the  same  effect  is, 
in  theory,  produced  as  if  a  series  of  roll- 


83 


ers  ?7i,  mj,  m^,  etc.,  upon  the  same  axis  as 
the  cylinder,  were  successively  applied  to 
the  surface  of  the  disk,  and  all  slipping, 
at  any  rate  from  this  cause,  is  avoided. 
The  actual  mechanism  which  has  not 
been  employed  for  a  planimeter  takes  a 
slightly  difterent  external  form  in  the 
harmonic  analyzer  of  Sir  W.  Thomson's 
tide-calculating  machine. 

The  devices  which  have  now  to  be  con- 
sidered as  solutions  of  the  problem  un- 
der consideration  by  the  first  method, 
are  used  in  connection  with  the  geomet- 
rical property  of  the  sphere  already  dis- 
cussed, and  upon  one  similar  to  it. 

Let  M  (Fig.  31)  be  the  plan  of  a  sphere 
rolling  along  the  line  OX,  carrying  with 
it,  by  a  frame  not  shown,  a  cylinder  {in) 
which  can  roll  about  it  so  as  to  come  into 
contact  at  anj  point  q  upon  its  horizon- 
tal great  circle.  Then  the  rotation  of 
the  cylinder  may  be  employed  exactly  in 
the  same  way  as  the  rotation  of  the  roller 
on  the  integrator  described  on  page  416? 
and  shown  by  Fig.  28  ;  but  in  the  pres- 
ent case,  instead  of  causing  the  roller  to 


84 


slip  over  the  surface,  the  rolHng  of  the 
cyhnder  is  practically  equivalent  to 
briuging  in  succession  a  series  of  roll- 
ers m,  ni^,  rii^,  etc.,  upon  one  axis  in  con- 
tact with  it. 

This  principle  has  been  employed  both 
by  Professor  Mitchelson  of  Cleveland, 
XJ.  S.,  and  Professor  Amsler.  The  mech- 
anism of  Professor  Mitchelson's  instru- 
ment is  shown  in  Fig.  32.  In  this  a  flex- 
ible steel  band  or  chain  F,  passing  round 
a  semi-circular  arc  D,  forces  the  cylinder 
C  to  roll  on  the  sphere  G.  The  cylinder 
is  carried  by  a  frame  E,  which  slides 
along  the  bar  A,  by  which  it  is  supported. 
The  mode  in  which  it  is  proposed  to  ap- 
ply it  to  the  ordinary  Amsler  planimeter 
is  shown  on  a  smaller  scale,  Fig.  32a, 
where  b  is  the  pole-arm,  a  the  radius 
bar,  and  t  the  center  of  rotation  of  the 
latter. 

Professor  Amsler's  planimeter  on  this 
principle  is  similar  to  the  foregoing,  ex- 
cept that  instead  of  being  carried  by  two 
guides  as  sleeves  by  a  bar,  the  cylinder 
frame   is    supported    on    rollers  from  a 


85 


Fig.  33 


86 


frame  above,  the  rolling  friction  on  the 
latter  being  less  than  that  of  the  cylinder 
on  the  sphere.  Thus,  the  cylinder  always 
moves  to  its  required  position.  The  mo- 
tion of  the  spherical  surface  is  obtained 
from  a  bevel- wheel  upon  its  axis,  which 
gears  with  a  larger  one  formed  upon  the 
edge  of  a  circular  stand  or  suj^port  of 
the  instrument. 

A  similar  principle  of  the  geometry  of 
the  sphere  has  also  been  employed  in  an 
instrument  suggested  in  a  p^iper  in  1855 
by  the  late  Professor  Clerk  Maxwell, 
when  an  undergraduate  at  Cambridge. 
Instead  of  the  cylinder  in  Fig.  31,  let  a 
sphere  m'  roll  around  on  the  sphere  (M), 
as  shown  in  Fig.  33.  Then,  from  the 
property  of  the  sphere,  which  is  proved 
at  length  in  the  above  paper,  the  tiu'ning 
of  the  sphere  m  about  its  axis  of  rota- 
tion xx^  relatively  to  the  turning  of  M 
along  OX,  is  proportional  to  the  tangent 
of  the  angle  a  in  the  figure.  In  the  other 
case  it  will  be  remembered  that  the  turn- 
ing of  the  cylinder  or  disk  was  propor- 
tional to  the  sine  of  the  same  angle.     By 


87 


suitable  means  the  "principle  can  be  em- 
ployed in  the  constiTiction  of  a  planim- 
eter.  Two  forms  of  such  planimeter 
are  shown  in  the  paper,  and  though  they 
are  both  in  the  form  of  the  linear  plan- 
imeter, and  are  scarcely  suitable  for  prac- 
tical application,  yet  the  matter  is  dealt 
with  in  a  way  worthy  of  the  inventor. 
It  is  evident  that  this  is  another  case  of 
bringing  the  eqiiivalent  of  a  series  of 
rollers  ;//,  m ^^  ?//,,  into  contact  with  the 
sphere,  thougli  these  are  no  longer  of  one 
size,  but  vary  from  a  diameter  zero  to  a 
diameter  of  the  size  of  that  of  the  sphere 

Coming  now  to  the  instruments  in 
which  the  alternative  device  adopted  for 
the  avoidance  of  slipping  is  by  bringing 
into  .contact  with  one  roller  different 
circles  of  the  disk  M,  or  of  its  equivalent. 

This  may  be  done  in  the  following 
way :  Instead  of  allowing  the  cylinder 
(m)  to  roll  on  the  sphere  M  (Fig.  31), 
and  so  to  change  the  radius  of  the  imag- 
inary rolling  circle  (whose  diameter  is 
qq')  on  which  it  rolls,  suppose   that  the 


88 


cylinder  is  kept  in  contact  as  shown  by 
the  dotted  lines,  and  the  axis  of  rotation 
zz'  of    the    sphere  is  tui'ned,   as,  for  in- 


(Z) 

stance,  would  happen  if  a  sphere  in  com- 
bination with  rollers  were  used  as  sug- 
gested for  an  anemometer  by  Mr.  Ven- 
tosa,  through  an  equivalent  amount,  i.  e., 
through  the  angle  a.     This  will  give  the 


89 


same  result  as  far  as  the  rotation  of  the 
cylinder  is  concerned,  but  with  an  im- 
portant difference.  The  cyhnder  (in  Fig. 
31)  or  sphere  (in  Fig.  33)  is  no  longer 
needed,  and  may  be  replaced  by  the  orig- 
inal measuring  roller,  whose  axis  has  a 
fixed  position  parallel  to  OX.  It  will  be 
seen  that  this  device  practically  amounts- 
to  bringing  different  circles  on  the  sphere 
M  into  contact  with  the  measuring  roller 
(ni),  with  the  great  advantage  that  ex- 
actly the  same  circle  on  the  sphere  M  is 
scarcely  likely  to  again  roll  in  contact 
with  the  roller  {)/i),  tliough  of  course  the 
radius  maj'  be  the  same.  This  method 
has  been  recently  proposed  by  the  author^ 
and  the  mode  of  carrying  it  out  without 
involving  slipping,  by  what  is  called  the 
"  sphere  and  roller  mechanism,"  which 
mechanism  has  been  explained  and  de- 
veloped at  length  in  a  paper  before  the 
Royal  Society.  It  need  here  be  only  re- 
marked that  the  planimeter  there  de- 
scribed, and  afterwards  exhibited  to  the 
British  Association  at  Montreal,  was  of 
the  linear  form,   and    of    little   practical 


90 

use  ;  but  the  author  has  since  completed 
a  polar  planimeter  and  exhibited  it  be- 
fore the  Koyal  Society. 

But  one  more  area  planimeter  re- 
mains to  be  mentioned,  aod  this  is  the 
one  invented  and  brought  before  the 
Physical  Society  by  ]\Ir.  C.  V.  Boys.  The 
principle  of  action  is  briefly  this  :  A 
wheel  or  roller,  which  is  not  supposed 
to  slip  sideways  on  the  diagram,  has  its 
plane  of  rotation  kept  always  at  an  angle 
a  to  the  axis  OX  of  the  figiu-e  to  be  inte- 
grated, such  that 

2/= ordinate  of  the  curve 
=rtan  axK 
where  K  is  a  constant,  and  y  is  the  ordi- 
nate with  resj^ect  to  OX  of  that  point  on 
the  curve  which  the  pointer  of  the  instru- 
ment is  at  the  same  instant  tracing.  If 
the  component  of  a  small  motion  of  the 
wheel  parallel  to  OX  is  a  a,  and  the  com- 
ponent of  the  same  movement  parallel 
to  OY  is  i\t. 

Then 


A^ 

A.!' 

=  tan  a  — 

y 

K 

A^  = 

-y  Axx 

1 
K' 

91 


or  the  distance  moved  by  the  wheel  par- 
allel to  the  axis  OY  becomes  the  measure 
of  an  element  of  area.  It  is  easy  to  see 
that  the  height  moved  by  the  wheel  be- 
comes a  direct  measure  of  the  area  of  the 
figure.  Various  examples  of  the  action 
of  this  planimeter,  called  by  the  inventor 
the  tangent  integrator,  are  given  by  Mr. 
Boys  ;  but  the  action  is  obviously  limited, 
and  an  investigation  of  the  theory  reveals 
the  fact  that  it  is  only  a  special  case  of 
the  general  problem,  not  only  of  the 
method  of  applying  circles  of  varying 
diameter  to  one  roller,  but  of  the  sphere 
and  roller  mechanism  itself.  This  will 
be  rendered  clearer  by  stating  that,  in 
order  to  employ  the  component  parallel 
to  OY,  the  roller  was  made  to  work 
against  a  cylinder,  which,  by  its  turning, 
acted  as  the  measuring  roller.  Evidently 
the  length  of  the  cylinder  limited  the 
travel  in  that  direction.  The  cylinder 
was  carried  bodily  along  in  the  direction 
of  its  axis  (corresponding  to  OX),  and 
made  to  effect  its  own  turning,  the 
amount  of  turning  varying  with  the  tan- 


92 

gent  of  inclination  of  the  wheel,  and  this 
was  sufficient  in  the  application  to  the 
steam-engine  integrator  to  be  hereafter 
described,  where  the  longitudinal  motion 
of  the  cylinder  could  be  made  propor- 
tional to  the  stroke.  Mr.  Boys  endeav- 
ored, by  various  means,  to  obtain  con- 
tinuous motion  in  both  directions,  one 
being  equivalent  to  bending  the  ends  of 
the  cylinder  round,  and  so  attempting  to 
solve  the  difficulty  by  what  he  has  termed 
a  "mechanical  smoke  ring."  The  author, 
however,  by  approaching  the  matter  from 
a  different  point  of  view,  designed  the 
sphere  and  roller  integrator,  which  is 
nothing  more  or  less  than  the  inversion 
of  the  mechanism  of  Mr.  Boys.  In  this 
the  roller  of  Mr.  Boys  is  replaced  by  the 
sphere,  and  instead  of  the  two  motions, 
one  of  the  cylinder  about  its  axis,  and 
one  of  the  cylinder  longitudinally,  the 
two  rollers  are  used.  It  may  be  easily 
shown  that  the  turning  of  the  plane  of 
rotation  of  the  roller  of  the  tangent  in- 
tegrator is  equivalent  to  changing  the 
axis  of  the  sphere  in  the  sphere  and 
roller  integrator. 


93 


Moment  Planimeters. 
The  moment  of  an  area,  and  its  mo- 
ment of  inertia  about  a  given  line,  may 
be  obtained  mechanically  upon  similar 
principles  to  those  by  which  a  simple 
area  was  obtained.  If  ABCDE,  Fig.  1, 
be  the  figure  whose  moment  of  area  and 
moment  of  inertia  are  required  about  any 
line  OX ;  then,  taking  any  element  of 
area  AB,  if  //  =  height  of  upper  portion 
SB,  then  the  moment  of  area  of  the 
element  SB  about  OX 

is  mz=area  of  SBx~- 

Similarly,  the  moment  of  inertia   of  the 
element  is 

1=^1/;  AX. 

The  sum  of  an  infinite  number  of 
such  expressions  as  these,  when  A  x  be- 
come infinitely  small,  gives  respectively 
the  moment  of  area  and  the  moment  of 


94 


inertia  of  the  whole  figure  according  to 
the  expressions. 

Moment  of  area    =M  =  ^ /'yVZa;, 
Moment  of  inertia = I    ^=z^f  (fdx. 

Now,  there  are  two  possible  ways  of 
obtaining  these  results  mechanically. 
One  of  these  ways  is  by  applying  for  the 
purpose  the  suggestion  made  by  Sir  Wil- 
liam Thomson  in  connection  with  the 
disk  globe  and  cylinder  integrator  of 
Professor  James  Thomson,  of  using  a 
train  of  such  mechanisms  to  obtain  the 
integration  of  a  simple  linear  differen- 
tial equation.  By  certain  simple  arrange- 
ments. 

The  first  mechanism  would  gi\efydx, 
"     second         "  "         "-  fy^dx. 

"     third  "  "         "  fy'dx. 

This  method  need  not  be  further  con- 
sidered here,  since,  so  far  as  the  author 
is  aware,  it  has  never  been  carried  into 
actual  practice.  It  maybe,  however,  said 
that  the  mechanical  difficulties  in  the  way 
of  causing  the  measuring  wheel  or  roller 
of  the  first  mechanism  to  actuate  the  sec- 
ond, and  the  roller  of  the  second  to  actu- 


95 


ate  the  third,  without  introducing  serious 
error,  are  not  easy  to  overcome,  and  re- 
quire a  very  easily  working  piece  of  ap- 
paratus. The  author  has  discussed  the 
apphcations  of  the  sphere  and  integrator 
for  tlie  purpose,  in  a  paper  to  the  Koyal 
Society. 

The  other  principle  is  to  cause  the 
measuring  roller  to  be  directly  turned 
at  a  rate  which  is  made  to  vary,  not  as 
in  the  simple  planimeter  with  the  value 
of  the  ordinate  (y),  but  with  its  second 
or  third  power.  Though  no  method  of 
directly  doing  this  has  apparently  yet 
been  suggested,  yet  the  same  result  is 
practically  effected  by  the  beautiful  appli- 
cation of  a  mathematical  principle  in  the 
"moment  integrator"  of  Professor  Ams- 
ler. 

Let  the  pole-arm  CB  (Fig.  34)  be  at- 
tached to  a  toothed  segment  {z^),  one 
portion  of  which  gears  with  a  toothed 
wheel  ^2,  the  radius  being  as  2  to  1.  Let 
the  center  C  of  s  be  carried  along  OX, 
while  the  center  of  C„  of  s^  is  carried 
along  a  line  ox  parallel  to  OX.     Let  m^ 


96 


97 


be  a  roller  acting  in  every  way  as  the 
measuring  roller  of  the  Amsler  planim- 
eter,  whose  axis  is  carried  in  the  plane  of 
the  wheel  z.^,  its  direction  passing  through 
the  center  C^.  When  the  pole-arm  coin- 
cides with  OX,  let  the  plane  of  rotation 
of  the  roller  m^  be  parallel  to  OX,  and 
its  axis  parallel  to  OY.  When  the  pole- 
arm  is  turned  through  an  angle  SCB=:a, 
the  angular  motion  of  the  wheel  z^  is 
twice  that  of  the  arm  ;  thus  the  roller  m^ 
takes  the  position  shown  in  the  figure. 

This  is  so  because  ^  — 


/  ^  motion  of  :c^ 

radius  2,  _2 
radius  z^     !> 

.•./Kc-/=2a. 


Suppose  the  pointer  p  to  move  through 
the  width  of  the  element  SB  at  a  height 
=:y,  and  with  it  z,  and  2.^,  the  roller  m^ 
being  in  contact  with  the  diagram  sur- 
face. Then,  by  what  was  proved  in  the 
case  of  the  Amsler  planimeter,  and  adopt- 
ing the  same  notation. 


98 


Turning  of  yn^  _Jl7trn^_  lc„, 

Motion  of  translation  of  t)}^        Ax       c^K 

=cos  2a 
=  1-2  sin' a, 

but       — ^  =  =(^r=sin  a  (where  CB=R  ), 

27r7vi,     -,0-2        -,       2       , 
AX  R/-^^' 


or 


."'  =  (2Fr)^*^-(2^'y^^- 


When  the  complete  travel  of  the  curve 
has  been  made,  the  sum  of  a  series  of 
quantities  similar  to  the  first  becomes 
zero ;    so  that,  by  making  the  constant 

I — rr-gl  equal  ^,  the  reading  of  the  roller 

gives  the  value — 

or  the  moment    of   area    of    the    figure 
BDEA. 

For  the  moment  of  inertia  the  segment 
of  2,  is  used,  the  radius  of  which  is  three 
times  that  of  another  wheel  z^,  with 
which  it  gears.  The  action  of  a  roller  771^, 
carried  by  the  wheel  z^,  is  exactly  the 
same  as  that  of  m^,  except  that  its  angu- 


99 


lar  motion  is  three  times  as  great  as  the 
pole-arm  CB,  instead  of  twice  as  great, 
as  in  the  case  of  the  other  roller. 

By  reasoning  similar  to  that  already 
adoj)ted,  and  taking  the  plane  of  rotation 
of  rn^  perpendicular  to  OX  in  its  initial 
position,  instead  of,  as  in  the  former  case, 
parallel  tolt — 

travel  of  rti^  Inrn^     l'c„ 


motion  of  translation 

of 

m.^        A  X 

-c,k' 

'  =  sin  3a 

=  3  sin  a- 

-4 

sin^  a. 

SB 
CB" 

"K" 

=  sin  «, 

Therefore ?  = 

:3  sin  0 

[  —  4  sin'a  = 

-B^- 

n.  .-/     ^    U 

A,      0« 

(.       *           \„. 

A    --.• 

which,  when  the  pointer  is  taken  around 
the  curve,  gives,  wuth  suitable  values  of 
the  constants, 

=  area  of  BDE A.— moment  of  in- 
ertia of  BDEA 
=^  A  -I. 

or  1=  A  — ^<,. 


100 

The  instrument,  Figs.  35,  36,  has  an 
area  planimeter  attached  to  it,  so  that,  by 
reading  the  rollers  m^  and  m^^  and  sub- 
tracting the  results,  the  moment  of  iner- 
tia is  obtained. 

The  details  of  the  moment  planimeter 
shown  (Figs.  35  and  36)  are  easily  ex- 
plained. A  guide  PQ  of  steel  has  a 
groove  gg,  which  is  placed  parallel  to  the 
axis  OX  by  means  of  the  gauges  G,  one, 
as  shown,  being  at  each  end,  which  are 
adjusted  with  their  points  q  upon  the 
line  OX.  The  rollers  KR  run  in  the 
grooves  gg,  and  support  a  frame  FF, 
which  carries,  by  means  of  an  axle  J  J, 
the  frame  EE.  This  frame  supports  the 
toothed  segment  2,,  and  the  two  toothed 
wheels  z^,  2^,  upon  vertical  axis.  The 
former  between  centers,  one  of  which  is 
shown,  Fig.  36,  ^,  the  latter  by  steel  axles 
within  the  column  ij,^.  The  pole-arm 
carries,  in  addition  to  the  pointer  p  at 
the  end,  the  roller  m^  with  its  dial  A,, 
forming  an  ordinary  planimeter,  and  is 
itself  carried  on  the  centers  s^s^.  The 
two  other  rollers  and  dials  are  shown  as 


101 


102 


rn^rn^  and  hji^  respectively.  The  weight 
W  serves  to  balance  the  instrument,  so 
as  to  avoid  undue  pressure  on  the  paper, 
and  the  motion  is  so  smooth  as  to  en- 
able a  curve  to  be  traced  with  the  great- 
est ease  and  accuracy. 

Attention  has  been  called  by  the  late 
Dr.  Merrifield  and  others  to  the  valuable 
applications  of  this  instrument  for  pur- 
poses of  naval  architecture,  but  so  far  as 
the  author  is  aware,  no  account  has  been 
given  in  this  country  of  its  applications 
in  civil  engineering,  as  proposed  by  Pro- 
fessor Amsler.  The  following  brief  ac- 
count of  the  methods  in  the  case  of  cal- 
culating the  contents  of  embankments, 
cuttings,  etc.,  is  therefore  given  from  an 
abstract  for  which  the  autlior  is  indebted 
to  the  kindness  of  Dr.  A.  Amsler : 

Let  Fig.  37  be  the  plan  of  a  portion  of 
an  embankment  or  cutting,  the  character 
of  which  is  supposed  to  be  the  same 
throughout,  viz.,  of  uniform  width  of 
roadway,  and  uniform  side-slopes,  the 
surface  of  the  ground,  the  gradient,  and 
the  horizontal  curvature  of  the  roadway, 


103 

being  restricted  in  no  way.  AA'  repre- 
sents the  center  line  of  the  railway ; 
B^B/  and  B^B/  its  two  borders ;  C^C/ 
and  C^C/  the  intersections  of  the  side- 
slopes  with  the  surface  of  the  ground. 
Suppose  now  the  embankment  or  cutting 

Fig.  sr 


to  be  divided  into  thin  layers  by  vertical 
planes,  perpendicular  to  the  center  line 
AA'  of  the  roadway ;  PQ  and  PjQj  may 
be  the  intersections  of  two  adjacent 
planes  with  the  plane  of  the  diagram. 

Then  if     p=area  of  section  PQ, 

As=interval  between  PQ  and 
and  P,Q,  measured  upon 
the  center  of  gravity  of 
the  section. 


104 

Total   volume   of   embankment  is  (from 
one  of  the  properties  of  Gnldinus) — 

Y=fpds, 
the   integral  extending   over   the  whole 
length   of   the   embankment  under  con- 
sideration. 

There  are  three  cases  dealt  with  in  the 


Paper  of  Professor  Amsler,  correspond- 
ing to  the  three  forms  of  sections,  I,  II, 
or  III,  Fig.  38. 

The  first  of  these,  I,  is  simple  enough, 
since  the  center  of  gravity  of  the  section 
always  coincides  in  plan  with  the  center 
line  of  the  roadway,  and  the  plan  of  oper- 
ation is  as  follows  : 

Let  Fig.  39  represent  a  longitudinal 
section  of  a  portion  of  the  embankment 
of   uniform   gradient,   developed   into   a 


105 


106 

23lane;  the  straight  hne  E'EE" represents 
the  top  of  the  embankment ;  G'GG"  the 
profile  of  the  ground  ;  the  straight  Hne 
A'AA'^  which  is  parallel  to  E'EE",  is  the 
locus  of  the  imaginary  vertex  of  the 
trapezoidal  cross-sections.  The  level 
line  MN  is  the  line  to  which  the  offsets 
of  the  profile  of  the  surface  of  the  ground 
refer.  BG  shows  the  intersection  of  a 
vertical  cross-section  with  the  figure,  and 
AG  the  intersection  of  a  plane  perpen- 
dicular to  the  top  of  the  embankment 
(and  also  to  the  line  A'AA")  with  the 
figure. 

Let  i=  Z  AGB=:  /  MNA=:gradient ; 

y=:AG  =di stance  of 
vertex  to  bot- 
tom  of  em- 
bankment ; 
2/o=AE  ^distance  of 
vertex  to  top 
of  embank- 
ment ; 
2  /i= angle  at  vertex  at  A. 

It  may  be  easily  proved    the  area  of 
the  section  made  by  the  plane  AEG  is — 
p^{kG^'--kW)  ian/i={y'-^;)  tan/i 


107 

but  since  Y=fpdx 

Therefore  volume  =  tan  ftf  {y'^—y^)  <^^^- 

And  thus,  if  (i  is  known,  the  volume  of 

the  portion  E'E"  G"G'  (Fig.  39)  is  easily 

found  with    the   mechanical    integrator, 

thus: 

Take  A'AA"  as  the  axis  of  moments, 
and  adjust  the  rail  of  the  instrument  so 
as  to  be  parallel  to  it.  Start  the  pointer 
anywhere  on  the  shaded  figure,  and  trace 
round  it ;  the  travel  of  the  roller  m^  being 
denoted  by  M,  the  scale  of  the  drawing 
longitudinally  being : 

l"=m  feet, 
and  vertically  l"=7i  feet; 

then  volume  =:Y  =  20    mn^   tan 

/ixM  cubic  feet. 

It  only  remains  to  insert  a  known 
value  for  tan  fi,  which  is  easily  done, 
thus  : 

Let  Fig.  40  be  a  perspective  view  of 
the  sections  AEG  and  BG  (Fig.  39), 
where : 

C  G 

Then  from  the  diagram     ^-J^=tan   ; 


108 


also 


BG 


=  tan  a 


Therefore 


or 


^  AG 

and  — —  =cos  i. 

±>(jr 

tan  a 


tan  /?=■ 
\  cos  ^/ 


cos  I 
,tan  a> 


ns.40 


where  Z  a  and  Z  ^'  are  known  constants. 
To  complete  the 
-  calculations  for  the 
whole  route  separate 
portions  are  taken, 
with  the  various  pro- 
posed gradients. 

The    above    formula 
is  exact  for  the  integ- 
rator   shown    in    Fig. 
36,    as    arranged    for 
English     measures,     a 
complete  revolution  of 
the    measuring    roller 
being  taken  as  a  unit 
of  reading. 
It    is    to    be    noted  that   nothing    is 
supposed   as    to    the    curvature   of    the 
center  line  of   the  roadway  horizontally, 
as   it   is   supposed   to   be   developed   in 


109 

the  figure.  Also,  that  the  aggregate 
error  arising  from  the  assumptions  that 
the  cross-sections  are  exact  trapezoids 
will  in  most  cases  be  verj^  slight,  on  ac- 
count of  the  errors  in  cuttings  and  those 
in  embankments  partly  compensating  for 
each  other,  in  addition  to  the  cutting  and 
filling  in  each  section,  as  shown  in  Fig. 
41,  where  the  small  triangular  portion  in 
dotted  lines  C'DH  represents  the  amount 
taken  off  the  former,  and  added  to  the 
latter. 

Alterations  of  the  proposed  roadway, 
otherwise  involving  tedious  calculations, 
simply  necessitate  an  alteration  in  the 
line  A'AA",  and  a  repetition  of  the  me- 
chanical work  of  the  integrator,  but  need 
no  fresh  diagram.  In  preparing  the 
drawing,  allowance  should  be  made  for 
ditches  along  the  roadway  in  cuttings, 
which  is  easily  done,  as  shown  in  Fig.  42, 
where  B^B,,  which  equalizes  the  amounts 
taken  and  left,  must  be  considered  as  the 
roadway  line.  In  the  case  shown  in  Fig. 
43,  the  excess  of  the  embanking  over 
the  cutting  is  approximately  equal  to  the 


110 


Ill 


112 

layer  above  the  dotted  line  CjC^.  The 
contents  of  this  layer  could  be  measured 
either  by  considering  it  as  an  embank- 
ment, and  treating  it  as  such,  or  by  the 
simpler — and  for  a  first  estimate  suf- 
ficiently accurate  method — of  assuming 
its  section  to  be  a  parallelogram.  The 
area  of  the  shaded  portion  (Fig.  43)  is 
then  simply  to  be  measured,  and  the  re- 
sult, multiplied  by  the  length  of  the  road, 
gives  the  required  contents.  The  sup- 
position that  the  slopes  CD  and  CD'  are 
the  same  is  also  sufficiently  accurate. 

The  foregoing  is  the  first  method  de- 
scribed by  Professor  Amsler,  and  is 
extremely  simple,  but  obviously  only 
approximately  accurate.  The  two  other 
methods  are  capable  of  giving  very 
accurate  results,  and  are  dealt  with  by 
him  at  considerable  length.  Only  a  short 
account  of  them  will  be  given  here. 

The  first  thing  to  be  noted  is  that,  as  a 
rule,  the  center  of  gravity  of  tbe  section 
will  not  really  coincide  in  plan  with  the 
center  line  of  the  roadway,  but  will  curve 
at  the  line  SPP'S',  Fig.  44,  AA'  being  the 


I' 
i'M 


114 

true  center'line.  Thus,  in  the  expression 
fpds,  the  value  of  ds  does  not  coincide 
with  dx^  as  hitherto  assumed.  From  the 
figure  it  is  seen  that : 

A  s_K  +  e 

where  R  is  the  radius  of  curvature  of  the 
center  Hue. 


Therefore       As=Aa3l  +  :^j 


'pedx 


or         V=:  J  pds = J  pdx,  +  J  -^5—1 

and  this  expression  must  be  used. 

The  first  of  the  two  methods  assumes 
the  base  of  section  to  be  inclined,  but 
not  broken  (Fig.  38,  II),  and  the  side- 
slopes,  gradient,  and  radius  of  a  given 
portion  to  be  constant.  A  diagram  is 
prepared,  as  shown  in  Fig.  45,  in  which 
the  dotted  lines  now  represent  intersec- 
tion of  the  sides  of  embankment  with  the 
surface  of  the  ground,  which  do  not,  as 
before,  coincide  with  the  contour  of  the 
center  line. 


115 
From  this  figure      y^=AI!> 

2/?=  Z  at  vertex  A. 


P^ig.  45 


Then,  by  similar   reasoning    to    that 
previously  employed,  it  may  be  proved 
that : 
Area  of  element  section 


and 


^p=y^i/,  (2/,-?/J 


tanVi 


116 
.♦.  V^tan  ft  /  {y,y-y:)  cU^^^^ 

By  a  simple  transformation  this  ex- 
pression is  brought  into  such  a  form  as 
to  allow  of  mechanical  integration.  The 
final  formula  being : 

where 

^ ={/(}/' -y:)<i^-hr(y  :-y:)  dx- 

hf{y-y,)'dx- 

Another  simple  diagram  has  to  be  pre- 
pared, and  by  means  of  three  operations 
of  the  integrator,  the  values  of  U  and  Y 
are  given  thus : 

U  =  20x??i?i'  (^1  +  ^0  —  ^3) 
-^=mn'  [320  {u^-u^-u;)-100  {lo- 

where  m  and  n  have  the  significations 
formerly  explained,  and  u,  v,  and  lo  are 
the  respective  readings  of  the  area,  mo- 


117     • 

ment  of  area,  and  moment  of  inertia 
rollers  in  each  of  the  three  operations. 
Considering  the  great  amount  of  calcula- 
tion thus  saved,  and  the  accurate  nature 
of  the  results,  this  second  method,  al- 
though involving  rather  more  labor  than 
the  first,  is  a  very  important  one. 

The  third  method,  which  deals  with  the 
broken  base,  is  much  the  same  in  princi- 
ple as  the  second,  but  the  expressions 
become  more  complicated,  whilst  six 
readings  of  the  measuring  rollers  are  in- 
volved. The  case  of  an  embankment, 
consisting  partly  of  a  cutting,  is  com- 
pletely and  accurately  worked  out  by  this 
method. 

The  Paper  of  Professor  Amsler  con- 
cludes with  an  example  of  the  application 
of  his  integrator  to  the  problem  of  the 
strength  of  a  girder. 


118 


Continuous  Integrators. 

Any  piece  of  mechanism  which  contin- 
uously adds  np  results  may  be  regarded 
as  a  continuous  integrator.  Of  such  in- 
struments, revolution  counters  as  em- 
ployed in  meters  of  various  kinds,  form 
the  simplest  example,  and  correspond  in 
action  to  the  devices  already  described, 
by  which  the  linear  measurement  of  a 
boundary  is  performed.  These  will  not 
be  further  referred  to,  and  it  is  only  nec- 
essary to  consider  those  computing 
mechanisms  which,  dealing  with  the  re- 
sult of  two  simple  imit  measurements, 
correspond  in  principle  to  area  planim- 
eters. 

It  appears  that  Poncelet,  before  the 
year  1838,  suggested  the  employment  of 
a  continuous  integrator  for  computing 
the  two  factors  in  dynamometrical  meas- 
urements. This  was  described  by  Morin 
in  1838,  as  applied  in  his  "  compteur " 
for  registering  the  work  done  by  a  team 
of  horses,  dragging  a  loaded  carriage  at 


119 

any  given  velocity  over  any  length  of 
road.  The  princii^le  employed  was  that 
of  the  disk  and  roller,  the  use  of  which 
as  already  shown,  had  been  suggested 
for  application  in  a  planimeter  more  than 
twenty  years  before.  In  the  case  in 
question,  the  disk  was  turned  by  an  end- 
less cord  or  band  from  one  of  the  wheels 
of  the  carriage,  while  the  position  of  the 
roller  on  the  disk  was  caused  to  vary 
with  the  tractive  force,  and  its  reading 
thus  gave  the  product  of  force  and  space, 
or  the  actual  work  done. 

In  1840,  a  Committee  of  the  British 
Association,  consisting  of  Professor 
Mosely,  Mr.  Enys,  and  Mr.  Hodgkinson, 
was  appointed  to  procure  the  dynamo- 
metrical  apparatus  of  Mr.  Poncelet,  and 
to  obtain  a  series  of  experiments  on  the 
duty  of  steam-engines  by  means  of  that 
apparatus,  the  sum  of  £100  being  placed 
at  their  disposal  for  the  purpose.  The 
report  of  this  committee,  in  1841,  de- 
scribes at  length  the  "constant  indi- 
cator "  of  Professor  Mosely,  which  was 
in  reality  a  continuous  steam-engine  in- 


120 

tegrator.  It  was  entirely  a  new  instru- 
ment, except  that  the  principle  of  the 
disk  and  roller  was  employed,  and  also 
the  traction  springs  of  General  Morin. 
The  pressure  of  steam  was  allowed  to 
act  upon  a  piston  so  as  to  vary  the  posi- 
tion of  the  measuring  roller,  while  motion 
was  given  to  the  surface  of  revolution  by 
means  of  the  stroke  of  the  engine.  The 
surface  of  revolution  was  a  cone,  which 
was  substituted  for  a  disk,  as  by  this  ar- 
rangement the  rapidity  of  the  changes  of 
velocity  due  to  corresponding  changes  in 
the  position  of  the  integrating  wheel  is 
diminished  in  the  same  proportion  in 
which  the  sine  of  one-half  the  angle  of 
the  cone  is  less  than  unity.  The  force 
necessary  to  drive  the  integrating  wheel 
is  diminished  in  the  same  proportion,  and 
therefore  the  chance  of  an  error  arising 
from  the  slipping  of  the  edge  of  the  in- 
tegrating wheel  on  the  surface  which 
gives  it  motion.  The  reports  of  the 
committee  in  1842,  1843,  and  1844  (which 
was  joined  in  the  first  of  these  years  by 
Dr.  Pole),  show  that  the  action  of  the 


121 


above  instrument  was,  as  far  as  could  be 
determined  in  its  application  to  a  single- 
acting  Cornish  engine,  very  satisfactory, 
but  apparently  no  mention  has  been  made 
of  it,  or  results  obtained  from  it,  in  any 
succeeding  report. 

Various  other  steam-engine  integrators 
have  been  brought  forward  since  then, 
most  of  them  acting  upon  the  same  prin- 
ciples ;  amongst  these  is  the  recent  power- 
meter  of  Messrs.  Ashton  and  Story,  which 
is  described  at  considerable  length  in  the 
American  edition  of  Weisbach's  Mechan- 
ics as  apparently  something  new.  It  is, 
however,  the  same  instrument  as  Mose- 
ley's  integrator,  except  that  it  employs  a 
spiral  instead  of  a  straight  spring,  and 
returns  to  the  use  of  the  flat  disk. 

The  disk  and  roller  has  also  been  since 
the  time  of  Morin  applied  in  many  dy- 
namometers, which  thus  become  really 
"  ergometers  "  or  "  work  "  measurers.  In 
a  series  of  articles  which  recently  ap- 
peared in  La  Lumiere  Electrique  those 
of  Hirn,  Megy,  Bourry,  and  Darwin,  are 
described  as  having  a  "  totalizer ''  or  in- 


122 

tegrator  of  this  kind  attached  to  them. 
The  position  of  the  measuring  roller 
varies  with  the  force  exerted  or  trans- 
mitted, and  the  motion  of  the  disk  with 
the  revolutions  of  the  motor  or  machine. 
Thus  by  suitable  counting  apparatus  the 
continuous  product  is  given  of  force  and 
space,  or  work  done. 

A  cone  has  sometimes  been  used  in- 
stead of  the  disk  with  dynamometers,  as 
in  that  of  Baldwin  and  Eickemeyer,  used 
at  the  Centennial  Exhibition  in  1876  for 
testing  mowing  machines,  in  which,  in- 
stead of  a  measuring  roller,  a  cylinder  is 
placed  with  its  axis  parallel  to  one  side  of 
the  cone,  while  an  endless  band  of  round 
cord  is  rolled  along  between  the  two  sur- 
faces, so  as  to  transmit  the  varying  mo- 
tion of  circles  of  different  radius  on  the 
cone.  Fig.  46  shows  the  apparatus  of 
Mr.  Koury  applied  to  a  transmission  dy- 
namometer. In  this  the  force  is  trans- 
mitted through  a  differential  train  of 
three  bevel  wheels,  the  middle  one  of 
which  is  attached  to  the  spindle  (S)» 
which  supports  the  weight  (W),  and  is 


123 

suspended  from  the  main  shaft  by  the 
joint  at  (k),  about  which  the  whole  can 
turn.     Thus,  the  deviation  of  the  weight 


from  the  vertical  (as  shown  by  the  dotted 
lines)  changes  with  the  force.  The 
change  of  position  of  the  spindle  (S) 
causes  a  band  {l>)  to  move  along  the  sur- 


124 

faces  of  revolution  RR,,  the  upper  one,  R, 
being  turned  from  the  shaft  by  the  spin- 
dle (ZZ).  It  is  to  be  noted  that  the  dis- 
tance of  the  band  (b)  from  its  zero  posi- 
tion is  not  directly  proportional  to  the 
force  represented  by  the  change  of  posi- 
tion of  the  weight,  and,  therefore,  the 
surfaces  must  be  formed  with  a  certain 
curve,  found  by  construction,  in  order 
that  the  dial  and  counting  apparatus  at 
D  may  correctly  give  the  product  of  the 
two  variables,  force  and  space,  and  so 
the  work  transmitted  through  the  dy- 
namometer. 

It  cannot  be  said  that  continuous  in- 
tegrators of  this  kind  are  at  present  prac- 
tically employed  to  any  great  extent. 
There  are  probably  two  reasons  for  this. 
One  is  the  want  of  durable  and  reliable 
instruments.  The  other,  the  question  as 
to  how  much,  and  to  what  degree  they 
are  really  needed. 

With  regard  to  the  first  of  these,  it  is 
evident  that  in  all  the  arrangements 
hitherto  considered  (with  the  exception 
of  Baldwin  and  Eickemeyer  device)  there 


125 


Fia-.  47 


is  that  slipping  of  surfaces  in  contact, 

which,  though  of  little  effect  as  far  as 
wear  goes  in  the 
limited  operations 
of  a  planimeter, 
becomes  a  very  se- 
rious considera- 
tion when  contin- 
uous action  is  re- 
quired to  be  main- 
tained. The  only 
integrator  of  the 
second,  or  non- 
slipping  class, 
which,  as  far  as 
the  author  is 
aware,  has  yet 
been  practically 
applied,  is  the 
"  power- meter  " 
of  Mr.  Vernon 
Boys.  This  instru- 
ment is  shown  in 
Figs.  47  and  48, 
and  acts  upon  the 

same  principle  as  Mr.  Boys'  integrator. 

The   piston   C,    subject   to   the   varying 


126 

pressure  in  the  engine-cylinders,  with 
which  the  barrel  A  is  connected  by 
the  connections  at  B  and  B',  is  moved 
up  and  down  against  or  with  the  ten- 
sion of  the  spring  D ;  its  rod  acting  on 
the  arm  g  causes  the  plane  of  rotation 
of  the  roller  G  to  take  positions  more  or 
less  inclined  to  the  axis  of  the  cylinder 
H.  This  cylinder  H  is  moved  to  and 
fro  with  the  stroke  of  the  engine  by 
means  of  the  cord  L,  Fig.  48,  and  the 
roller  G  bemg  in  frictional  contact  with 
it  causes  it  to  turn  round  to  a  greater  or 
less  extent,  according  as  the  plane  of  G 
is  more  or  less  inclined  to  the  axis  of  H. 
The  amount  of  its  revolution  is  registered 
by  the  counting  apparatus  in  I  (Fig.  48), 
to  which  the  axis  of  H  is  geared,  and  is 
thus  a  measure  of  the  power  of  the  en- 
gine, for  it  gives  the  product  of  the  tan- 
gent of  the  angle  to  which  G  is  inclined 
and  the  distance  moved  through  by  H, 
that  is  the  product  of  pressure  of  steam 
into  the  stroke  of  the  engine.  The  steam 
being  (as  originally  in  Moseley's  and  also 
in  subsequent  integrators)  supplied  both 


127 


ff 


Fig.  48 


^ty= 


frS 


I     I 


B 


-  128 

above  and  below  the  small  piston,  the  ab- 
solute pressure  is  given.  Thus,  in  the 
present  case,  as  the  change  of  pressure 
on  C  at  the  beginning  and  end  of  each 
stroke  causes  the  rod  of  g  to  be  alter- 
nately above  or  below  the  axis  of  H,  so 
the  motion  of  the  cylinder  to  and  fro  will 
always  cause  the  cylinder  H  to  turn  in 
one  direction,  and  thus  to  continuously 
integrate  the  work  done.  This  device 
only  enables  a  reciprocating  movement 
of  the  cylinder  H  to  be  made,  and  the 
author  has  already  mentioned  the  device 
of  the  sphere  and  rollers,  which  by  the 
inversion  of  the  higher  pair  of  Mr.  Boys, 
enables  continuous  motion  to  be  obtained, 
and  is  suitable  for  application  in  dyna- 
mometers, electric-motors,  and  other  jDur- 
poses. 

With  regard  to  the  want  of  such  instru- 
ments, a  very  strong  case  was  made  out 
by  the  committee,  already  mentioned,  in 
their  report  in  1841,  where  the  applica- 
tion of  a  continuous  integrator  to  steam- 
engines  was  alone  discussed.  The  appli- 
cation has  been  made  to  electric-motors, 


129 

and  in  trials  of  motors  and  machines 
generally,  and  there  is  little  doubt  if  con- 
tinuous integrators  combining  the  three 
qualities  of  durability,  accuracy  and 
cheapness  could  be  produced,  that  in 
these  days  of  increased  regard  for  meas- 
urement of  all  kinds,  there  would  be  a 
much  larger  and  increasing  application 
of  them. 


Limits  of  Accuracy  of  Integrators. 

In  all  calculating  machines,  accuracy  of 
the  result  must  be  the  question  of  first 
importance.  Assuming  the  theory  relied 
on  in  the  various  instruments  for  the 
mathematical  operation  to  be  correct,  the 
accuracy  depends  primarily  upon  the  me- 
chanical arrangements,  though  in  the 
case  of  planimeters  it  also  depends  upon 
the  skill  and  care  of  the  manipulator,  and 
involves  the  question  of  a  personal  er- 
ror. This  latter  point  need  not  be  con- 
sidered, partly  because  this  occurs  more 
or  less  in  all  results  obtained  by  observers, 
but  also  because  it  is  less  than  might  be 


130 

at  first  anticipated,  from  the  fact  that  in 
tracing  the  pointer  around  the  curve 
there  is  no  reason  why  the  error  due  to 
moving  it  on  one  side  should  exceed  that 
due  to  moving  it  on  the  other  side,  that 
is,  why  equal  errors  of  opposite  effect 
upon  the  final  reading  should  not  be  made. 

It  has  been  seen  that  the  action  of  all  in- 
tegrators, except  mere  revolution  count- 
ers, depends  upon  the  motion  of  the  meas- 
uring roller,  or  its  equivalent,  over  sur- 
faces of  various  forms,  therefore  the 
above-mentioned  mechanical  arrange- 
ments resolve  themselves  into  an  exam- 
ination of  the  nature  of  the  frictional 
contact  of  two  surfaces.  It  was  for  this 
reason  that  integrators  have  been  classi- 
fied according  to  the  nature  of  this  fric- 
tional contact,  and  it  now  remains  to  in- 
vestigate the  nature  of  this,  to  show  to 
what  the  classification  leads,  to  give  the 
direct  results  of  experiments  upon  the 
subject,  and  also  the  indirect  results 
obtained  from  the  instruments  them- 
selves. 

Planimeters  and  integrators  generally 
have  been  divided  into — 


131 

I,  Those  in  whicli  the  frictional  sur- 
faces shp  as  well  as  roll  over  each 
other. 

II.  Those  in    which   slipping   of    the 


Fis.  49 


surfaces  is  supposed  not  to  take  place. 

The  order  of  this  arrangement  was 
adopted  upon  historical  grounds,  and 
also  because  the  former  class  is  at  pres- 
ent by  far  the  most  important ;  but  it 
would   be    more  convenient,  upon  mere 


132 


grounds  of  mechanical  simplicity,  to  in- 
vert the  order. 

Let  AB  (Fig.  49)  be  the  plan  of  the 
measuring  roller.  Suppose  a  force  ap- 
plied in  the  direction  OX,  making  an 
angle  (a)  with  the  plan  of  the  axis  of 
AB. 

Let  <:?=!  OX = distance  through  which 
the  force  acts. 

1st.  Suppose  that  frame  which  carries 
the  measuring  roller  is  free  to  move  in 
any  direction  horizontally,  but  maintains 
the  plane  of  rotation  of  the  roller  verti- 
cal, then  the  application  of  a  force  along 
OX,  at  the  center  of  AB,  will  cause  it  to 
roll  along  the  line  coinciding  in  direction 
with  the  plan  of  the  center  line  AB  of 
the  roller,  that  is,  along  the  line  OY. 
This  will  always  be  the  case,  except 
when  this  force  is  applied  in  the  limiting 
case  in  the  direction  perpendicular  to 
the  plane  of  AB  {i.  e.,  when  a  =  0). 

Thus,  the  distance  in  this  case  traveled 
by  the  center  of  AB,  which  is  the  same 
as  the  path  rolled  by  it,  is 

sm  a 


133 

and  the  distance  moved  through  by  the 
center  at  right  angles  to  OX  is 
XY=^cot  a. 

The  latter  value  is  the  one  usually 
taken  or  recorded  by  the  instruments  at 
present  in  use,  but  depends  directly  upon 
the  former. 

Next,  suppose  the  frame  carrying  the 
roller  is  constrained  either  by  guides,  as 
in  the  linear  planimeter,  or  by  the  radius 
bar  of  the  polar  planimeter,  or  other- 
wise, to  move  in  the  direction  of  OX, 
that  is,  in  the  direction  in  which  the  force 
acts.  When  the  center  of  the  roller  has 
reached  the  point  X,  that  is,  when  the 
force  has  been  exerted  through  a  dis- 
tance OX, 

Then  0Z  =  distance  slipped  by  AB  = 
d  cos  a. 
XZ==:  distance  slipped  by   0B= 
d  sin  «. 

Upon  the  degree  of  accuracy  with 
which  the  above  conditions  are  fulfilled 
depends  the  correctness  of  the  working 
of  all  integrators ;  for  not  only  do  these 
two  cases  entirely"   cover  the   action    of 


134 

the  two  classes  of  planimeters,  and 
the  corresponding  continuous  integra- 
tors, but  one  of  the  limiting  cases  in 
each,  viz.,  that  in  which  the  force  acts  in 
the  plane  of  rotation  of  the  wheel  (when 
a=90°),  represents  the  conditions  under 
which  the  wheel  of  the  boundary  meas- 
urer or  opisometer  is  employed.  It  may 
be  therefore  said  that  the  theory  of  me- 
chanical action  of  integrators  is  based 
upon  one  or  other  of  the  following  as- 
sumptions, in  which  the  limiting  case 
(namely,  when  a=r90),  is  included. 

Class  I. — That  the  rollmg  of  the 
planimiter,  when  slipping  is  allowed,  is 

Nj^X^jfZsin  a. 

Glass  II. — That  no  slipping  takes 
place,  which  amounts  to  the  assertion 
that 


■'sm  a 


Nj  and  N^  being  the  readings  in  each 
case,  and  k^  and  h^  suitable  constants  for 
the  instruments. 

It  is  easy  to  see  that  the  first  of  these 


J 


135 

is  really  the  assumption  made  for  all  in- 
struments in  Class  I. ;  but  in  the  various 
instruments  in  Class  II.,  it  is  only  with 
the  i^lanimeter  of  Mr.  Boys  that  it  be- 
comes directly  obvious  that  the  above  as- 
sumption is  made.  With  the  others, 
though  it  is  less  evident,  nevertheless,  it 
will  be  found,  on  examination,  to  be 
equally  true  that  the  second  supposition 
is  really  made,  and  that  upon  its  truth 
the  correct  action  of  all  instruments  in 
the  second  class  depends.  The  forces 
acting  in  each  of  the  two  cases  must 
therefore  be  taken  into  consideration 
and  the  mechanics  of  the  problem  exam- 
ined. 

Proceeding  in  order  of  simplicity,  Class 
II.  will  be  examined  first. 

Let  AB  in  both  cases  (Fig.  50)  be  the 
plan  of  the  measuring  roller. 

Let  S= reaction  of  surface  upon  which 
AB  rolls,  that  is,  the  force 
with  which  it  is  kept  in  con- 
tact with  it ; 

/^= coefficient  of  friction  between 
roller  and  surface ; 


136 

P=OC==reaction  of  surface,  which 
must  be  brought  mto  action 
in  a  horizontal  direction  to 
cause  the  roller  to  turn  on 
its  axis. 

Class  11.  {Fig.  50).— Suppose  the 
frame  in  which  the  roller  is  carried  to  be 
free  to  move  in  any  direction  horizontal- 
ly, let  a  force  be  gradually  applied  at  the 
center  of  the  roller  AB  in  the  direction 
perpendicular  to  the  plane  of  rotation. 
This  will  produce  no  effect  as  long  as  it 
is  less  than  the  maximum  resistance, 
which  can  be  opposed  by  friction  be- 
tween the  edge  of  the  roller  and  the  sur- 
face upon  which  it  rests,  that  is,  as  long 
as 

K=:OD  (Fig.  50)  <S//. 

When  the  force  K  is  equal  to  S//,  and 
acts  within  the  angle  6,  the  roller  AB 
will  move  with  uniform  motion  along  the 
line  of  action  of  the  force  without  turn- 
ing. The  same  thing  will  hold  if,  instead 
of  the  force  acting  perpendicularly  to 
the  plane  of  rotation,  it  acts  at  some 
oblique  angle  to  it  not   exceeding  a  cer- 


187 


CLASS  2. 


138 


tain  value  measured  from  the  normal. 
The  limiting*  value  of  this  angle  depends 
on  the  resistance  of  the  friction  of  the  axle 
of  the  roller  AB  to  turning.  Let  this 
angle  be  (0),  and  draw  OE  perpendicular 
to  OC,  meeting  the  circle  draw^  with  O 
as  center  and  radius  OD  in  E,  and  let 

6^=angleE0D. 
When  the  line  of  action  of  the  force  falls 
without  the  angle  6,  as,  for  instance, 
when  it  takes  place  in  direction  OF,  the 
roller  will  still  slip  along  the  line  of  the 
force,  but  the  roller  will  now  also  turn. 
The  component  in  the  plane  of  rotation 
will  now,  however,  be  of  a  magnitude 
such  that  the  motion  of  rotation  of  the 
roller  is  no  longer  uniform.  Since  only- 
uniform  motion  is  being  considered  to 
take  place,  the  conclusion  is,  that  when 
the  force  acts  at  an  angle  greater  than  6 
to  the  axis  of  rotation  (^.  e.,  when  a>6) 
it  must  never  be  so  great  as  to  cause 
the  roller  AB  to  slip,  and  therefore  only 
a  motion  of  pure  rolling  can  take  place. 
By  proper  mechanical  devices  the  roller 
can  be  made  to  turn  very  easily,  and  angle 
be  kept  very  small. 


i 


139 

The  magnitude  of  the  force  which 
must  be  appHed  in  any  position  of  the 
roller  to  effect  this  motion,  is 

sin  a 
=Y  cosec  a, 

and  is  at  once  given  by  the  intercepts 
drawn  from  O  to  OE  in  the  construction, 
shown  in  Fig.  50,  for  any  other  value 
of  a. 

Class  I.  (Fig.  51) — Suppose  that  the 
frame  does  oppose  restraint,  and  that 
this  restraint  is  such  as  to  always  cause 
the  center  of  AB  to  move  in  the  direc- 
tion in  which  the  force  acts.  Let  OF 
(Fig.  51)  lie  in  this  direction,  making  the 
angle  a  with  the  axis  of  AB,  draw  EF 
perpendicular  to  OF  from  the  point  E, 
then  by  the  triangle  of  forces.  The  force 
required  to  move  AB  is 

P^  =  OF  =  Syucos  {a-6). 

The  reaction  which  is  supplied  by  the 
frame  is 

QrrEF  =  S/<sin(a-0). 

By  describing  a  semicircle  upon  OE  a 


140 


Fig.  51 


CLASS  1, 


141 


construction  is  at  once  given,  as  shown 
in  Fig.  51,  in  which,  by  drawing  radial 
lines  from  O,  the  value  of  P^  for  any 
angle  is  at  once    given   by  the  intercept. 

The  above  investigation  is  by  no  means 
a  complete  one,  for  this  would  require 
a  discussion  of  the  moments  acting,  but 
having  obtained  the  above  result  by  the 
more  complex  method,  the  author  con- 
sidered it  unnecessary  to  introduce  a 
more  detailed  jDroof  than  has  been 
given. 

Comparing  the  two  foregoing  cases  by 
means  of  the  diagrams  in  Figs.  50  and 
51,  Diagram  AB,  it  is  clear  that  the 
forces  acting  always  diifer,  except  in  the 
limiting  cases;  then 

If  a  =  dO°        P,=P,=F=OC; 

a=    0"  P^    is      <S//; 

P,   is    ^S//. 

In  order  to  examine  to  what  extent  the 
assumptions  hold  good  upon  which  the 
accuracy  of  integrators  rests,  the  exjDeri- 
mental  determination  of  the  following 
points  is  needed: 


142 

I.  Case  of  boundary  measures  and  lim- 
iting case  of  botli  classes  of  area  and 
moment  integrators. — The  conditions  of 
rolling  of  two  surfaces  when  the  force 
acts  in  dii'ection  of  the  plane  of  rotation. 

II.  Class  I.  of  area  and  moment  plan- 


'F^i<^.CM2 


imeters. — Whether  the  rolling  is  propor- 
tional to  the  slipping,  as  assumed. 

III.  Class  II.  of  area  and  moment  plan- 
imeters. — Whether  there  is  any  slipping 
in  this  case,  and  if  so,  to  what  extent. 

Case  J.  There  are  two  separate  prob- 
lems under  this  head;  one,  where  the 
planes  of  revolution  of  the  two  rolling 


143 

surfaces  coincide ;  and  the  other,  when 
they  do  not  coincide. 

Dr.  A.  Amsler  has  experimentally  in- 
vestigated the  former  thus :  Two  accu- 
rately turned  disks,  SS',  (Fig.  52),  each 
200  millimeters  in  diameter,  were  placed 
with  their  edges  in  frictional  contact,  so 
that  the  mark  shown  on  each  coincided 
at  the  point  of  contact.  By  means  of  the 
apparatus  (Fig.  53)  the  lower  one  S  w^as 
turned  by  means  of  a  band  (BB')  through 
1700  revolutions  one  way  (thus  turning 
the  upper  one  S,),  and  then  1700  revolu- 
tions in  the  opposite  direction.  The 
marks  which  should  have  coincided  if  no 
slij^ping  had  taken  place,  were  now  found 
to  be  as  in  Fig.  52,  the  marks  being  a 
distance  =  f„  =  0.05  millimeter   apart. 

The  relative  error  is  thus  =  ^     ^tt^,  = 

('^'^""^^  40,000,000-  This  experiment  is 
only  of  value  to  show  the  error  when  the 
w^heel  travels  back  by  turning  in  the  op- 
posite direction,  and  at  the  most,  shows 
that  the  error  is  nearly  the  same  in  both 


145 

directions,  aud  does  not  prove  anything 
with  regard  to  the  action  of  the  measur- 
ing roller  of  the  boundary  measurer; 
concerning  this  point,  observations  were 
wanted  upon  the  results  of  the  first  1700 
revolutions.  A  very  good  way  of  exam- 
ining the  point  would  be  to  note  the  diver- 
gence of  the  marks  in  x  revolutions  when 
S  drives  S,,  then  cause  S^  to  drive  S  back 
through  the  same  number  of  revolutions, 
and  it  would  be  seen  whether  the  diver- 
gence was  due  to  a  difference  in  the  per- 
iphery of  the  wheels,  or  to  a  slipping  of 
the  surface  of  contact.  During  Dr.  Ams- 
ler's  experiments  no  one  was  allowed  to 
enter  the  room,  in  order  to  avoid  altera- 
tion of  temperature.  He  found  that  the 
heat  radiated  from  the  human  body,  or 
even  from  a  lighted  candle  placed  at 
some  distance,  had  a  perceptible  influence 
on  the  result. 

Fig.  54  shows  a  most  important  case, 
in  which  the  plane  of  rotation  of  the  sur- 
faces in  contact  do  not  coincide  with  that 
of  the  measuring  roller  {m)  being  actu- 
ated by  the  disk  M.     Fig.  55  shows  Dr. 


146 

Amsler's  apparatus  for  exaroining  this 
case,  in  which,  if  the  edge  of  the  roller 
has  any  width,  there  must  be  slipping 
action,  even,  though  the  force  alwa^-s  acts 


Fig. 

o4 

\ 

7Jl          i 

\ 

/:>>-' — r~- ---. 

^ 

><^ 

v-^ 

y^                   1 

^^ 

\ 

^x 

/ 

\          ! 

\ 

\ 

\ 

/ 

\       ! 

\ 

1 

\      1 
\ 

M 

\ 
\ 

1 

\ 

1 

\ 

1 

\ 

\ 

1 

\ 

\ 

1 
1 

\ 

\ 

/ 

\ 

\ 

\ 

\     / 

/ 

N 

\.^ 

"^■"■-.. j 

^^'' 

in  the  direction  of  the  plane  of  rotation. 
The  roller  m  which  rests  on  the  upper 
surface  of  the  disk,  which  latter  has  its 
edge  divided,  and  is  in  juxtaposition  with 
a  vernier  (y).  The  axis  of  the  roller  is 
fixed,  and  its  edge  is  thus  kept  always 
vertically  under  a  microscope  (K).     The 


147 


:Fis.55 


148 


position  of  the  disk  is  noted,  and  it  is 
then  moved  forward  about  8  revolutions 
(or  exactly  2,900°),  which  gives  the  roller 
about  130  revolutions,  and  a  mark  is  ob- 
served on  the  latter.  Then  in  theory  the 
result  of  giving  8  more  revolutions  to  the 
disk  in  the  same  direction  should  be  to 
bring  the  same  mark  of  the  roller  under 
the  microscope,  Practically  the  succes- 
sive motions  of  the  disk  will  be  a  little 
different,  so  that  the  second  advance  of 
the  disk  will  not  be  exactly  the  same  as 
in  the  first  case.  The  same  mark  on  the 
roller  is,  however,  always  brought  under 
the  microscope,  and.  the  difference  in 
turning  of  the  disk  is  what  is  noted. 
In  the  following  table — 

^=: number  of  experiment, 
(p= angle  by  which  the  disk  differs 
from  last  reading, 

so  that  the  second  column  gives  the  po- 
sitions of  the  disk  at  the  end  of  succes- 
sive advances  in  which  the  roller  is  made 
to  take  130  complete  revolutions,  the 
third  column  shows  the  travel  of  disk  in 
minutes  (2,900°  having,  of  course,  to  be 
added    to    the   readings).      The    fourth 


149 

gives  the  difference  between  these  and  a 
mean  value.  The  last  gives  the  ratio  of 
these  differences  to  the  travel. 


T 
-1: 
1 

s 

1 

05 

of 

COt>  OOCOi>OOJ>0 

oooooooooo 
oooooooooo 

1  1  1  1  1  +  1 ++  1 

7 
-a- 
1 

1 

OO^OOQOCOWOOOOOGO 

•-  oooooooooo 

1  1  1 i  1 ++ 1  +  1 

of 

T 

1 

-SI 

^^^ssss^g^ 

^^^^^^^^^^^ 

o  * 

•.s> 
1 

§§g55^gi^g^S 

^  00  O^  rH  O  -t  CO  ^  O  o 

•^ 

OTH05C0Tj<oi:Di>X)OO 
1—1 

150 


y/m 

a. 

CC 

\A 


£ 


151 

Case  II.  To  test  the  results  when  a 
roller  partly  rolls  and  partly  slips,  Dr. 
Amsler  used  the  apparatus  shown  (Fig. 
56)  in  plan  and  elevation.  In  this  C  is  a 
carriage,  running  upon  four  wheels,  on 
the  base  (B),  which  has  parallel  grooves 
planed  in  it ;  the  travel  of  the  carriage 
being  limited  by  two  stops  at  the  ends. 
Upon  the  surface  of  C  the  measuring 
roller  {iii)  rests,  being  attached  to  a  plate 
A.  By  means  of  the  graduations  on  A 
the  axis  of  {iii)  can  be  set  at  any  required 
angle  with  reference  to  the  direction  of 
motion  of  the  carriage.  The  frame  sup- 
porting the  roller  is  carried  on  the  disk 
by  means  of  pivots,  so  as  to  allow  (m)  to 
rest  on  the  surface  of  C  with  the  constant 
pressure  of  its  weight. 

If  a  wangle  of  axis  of  {jii)  in  the  direc- 
tion of  the  motion  of  the  car- 
riage ; 

s= motion  of  the  carriage  ; 
^<  =  turning  of  the  roller; 

Then  w=s  sin  a  =  s  cos(  ^  — aj=s  cos  /i. 


152 

If  <7>= actual  reading  of  vernier  v 
and  (p=q)^  when  0  =  90  or  /i  =  0; 
then  u=zs  cos  (^  —  ^o)' 

iL=s  COS  {(p„  —  cp^),  etc. ; 
Then  tan  ",  '^"^  ■^,-".  '^»^  ■?,, 


COS  {(p,-Cp^)       COS   (^,-  ^)o 

In  the  following  Table,  which  repre- 
sents the  results  of  experiments  when 
the  disk  was  covered  with  a  surface  of 
pear-tree  wood,  carefully  polished  (paper 
being,  however,  found  to  afiord  almost 
as  good  results) : 

z=as  before,  the  number  of  the  experi- 
ment; 

Bi= angle  of  inclination  of  roller  for  ex- 
periment (/)  ; 

«/^= motion  of  roller  as  observed  for  ex- 
periment (e)  ; 

?^i= motion  of  roller  as  calculated. 


153 


1 

OrHC0Ti<.rHO-.---rHOTHO 

1      +    +   +   +   +   +              1        1 

^ 

-d 
o 

=3     . 
3    ""^ 

a  ^  ^  T->  r^  CI  y^ '^  a>  a  <^ 

"^  lO  r^  CO  IC  C^i  O  O  O  "^  tH 
CO  IC  *-■?  >     O  C5  t>  L-  C>  w  ^ 

1       ^' 

o 

ociCia:)OOOco-<*rt^THr-i 

,  1 

perim 
u'. 

ci  c->  1-1 1-  o  o  o  t-  o  o  L-r 
■"^  o  1-H  o  »o  T— 1  <o  o  c;  o  th 
O  ic  CT  c5  O  C:  I-  i?5  0>  O  rt^ 

c:ooxoo-,oOTi^'*TH,-H 

i 
1 

;     O  O  O  O  O  tS  O  CO  O  CO  O 
CO         CO         CO         JC 

■^ 

^   oocoooo^^cgoo 

II 

o     OOOC:COt--^OCQCOOi-H 
(MC^l-^TfCOCOOOOO 

^ 

Oi-KNCOTtOCOt'XCSO 

Class  III. — The  actual  conditions  of 
motion  when  a  force  smaller  than  the 
component  of  S/^,  acts  obliquely  to  the 
plane  of  rotation  of  the  measuring  roller, 
do  not  appear  to  have  been  made  the 
subject  of  direct   experiment.     It  is  ap- 


154 

parently  always  tacitly  assumed  that  no 
slipping  takes  place.  But  this  crucial 
point  cannot  be  thus  left  to  mere  con- 
jecture, and  the  author  has  designed  a 
method  of  carefully  testing  this,  which 
he  has  not  yet  been  able  to  properly 
carry  out.  From  a  few  rough  observa- 
tions, there  seems  little  doubt,  howevei, 
that  some  slipping  always  does  take 
place,  and  that  its  amount  is,  in  the 
limiting  cases,  by  no  means  inconsider- 
able. 

Lastly,  a  few  words  may  be  said  con- 
cerning the  work  of  Professor  W.  Tinter 
and  of  Professor  Lorber.  The  former 
has  examined  most  carefully  no  less  than 
nine  different  planimeters,  from  which  he 
concludes  that  the  different  angles  at 
which  the  measuring  roller  of  the  jDolar 
planimeters  has  little  effect  upon  the  re- 
sult, and  that,  taking  one  turn  of  the 
measuring  roller  as  fz=100  square  cm., 
the  average  error  in  the  reading  was 
only  from  ^0.00075  to  0.0013,  according 
as  the  center  of  rotation  was  without  or 
within  the  area  to  be    measured.      The 


155 

work  of  Professor  Lorber  is  so  exten- 
sive and  elaborate  that  it  is  impossible 
to  do  more  than  give  in  the  most  brief 
form  the  results  at  which  he  has  arrived 
after  many  thousands  of  experiments. 
He  concludes  that  error  in  the  reading  is 
always  represented  by  an  equation  of 
the  form — 

dn=K  '  -\-M\/ny 
(172=  the  error  in  the  reading, 
K    and    //   being    con- 
stants, 
where  ??=:the  reading  of  the  meas- 

uring roller; 

the  above  equation  gives  rise  to  one  of 
the  following  form  : 

where  r= actual    area  to  be    meas- 

ured, 

and  ^/F,i  =  tl]e  error  in  the  result  ex- 

pressed in  terms  of  the 
area. 

The  following  are  the  results  given  in 
his  latest  paper : 
Linear  planimeter  dF 

=  0.00081/+ 0.00087^F7 


156 

Polar  planimeter 

=  0.00126/  + 0.00022  VF/ 
Precision  polar  planimeter 

=  0.00069/+ 0.00018  ^jy' 
Freely  swinging  planimeter 

=  0.00060/"+ 0.00026y'jy 
Simple  plate  planimeter 

=  0.00056/+  0.00084^1/ 
Rolling  (Coradi)  planimeter 

=  0.0009/+ 0.0006  ^jy 

The  degree  of  accuracy  represented 
by  these  results  may  be  inferred  from 
the  fact  that  in  one  case  of  the  last  plan- 
imeter, when 

d¥         1 
/=100  the  relative  ^^'I'^^'^-p  ^fg-ggQ" 


157 


Discussion. 

Sir  Frederick  Bramwel],  President, 
said  that  the  paper  having  been  read  only 
in  abstract,  there  had  been  no  mention 
whatever  of  what  the  author  himself  had 
done.  The  members  would  no  doubt, 
under  the  circumstances,  allow  him  con- 
siderable latitude  in  personally  explain- 
ing the  apparatus  on  the  table.  He  would 
not,  however,  ask  them  to  defer  the  ex- 
pression of  their  thanks  to  Professor 
Shaw  for  his  valuable  paper  until  this 
explanation  was  given  ;  but  he  felt  sure 
that  after  this  was  done,  it  would  be  still 
more  clear  that  those  thanks  were  well 
deserved. 

Professor  H.  S.  Hele  Shaw  said  that 
the  paper  had  been  only  read  in  the  form 
of  a  brief  abstract  because  from  the  nature 
of  the  subject,  and  its  method  of  treat- 
ment, it  appeared  advisable  that  he  should 
personally  give  a  short  accouut  of  its 
contents.      The    engravings,    of    which 


158 


there  were  a  good  many,  could  not  be 
prepared  in  time  to  be  sent  to  those  who 
were  Kkely  to  take  part  in  the  discussion, 
and,  therefore,  he  would  explain  the 
principal  points  which  he  believed  to  be 
original,  and  which  he  hoped  would  be 
thoroughly  discussed.  He  would  take 
this  opportunity  of  thanking  Professor 
Amsler  for  kindly  lending  him  several 
instruments,  some  of  which  were  now 
shown  for  the  first  time  in  this  country, 
having  been  sent  from  Switzerland  for 
the  pur^DOse,  at  Professor  Amsler's  own 
expense.  He  also  wished  to  thank  Mr. 
C.  V.  Boys  for  lending  him  models  of  his 
tangent  integrators,  and  also  two  of  the 
actual  instruments  to  exhibit ;  and  his 
friend  and  colleague,  Mr.  C.  D.  Selman, 
for  several  valuable  suggestions  and  as- 
sistance in  the  preparation  of  some  of  the 
diagrams.  The  author  then  proceeded 
to  explain,  by  means  of  the  diagrams  on 
the  walls,  and  by  models  which  had  been 
constructed  on  a  large  scale,  the  princi- 
ples of  the  classification  adopted  in  the 
paper  and  the  various  instruments  exhib- 
ited. 


159 

Mr.  William  Anderson  (of  Erith)  ob- 
served that  he  had  had  considerable  ex- 
perience with  continuous  integrators  in 
measuring  work  done  by  agricultural 
implements.  There  was  a  good  deal  to 
be  said  about  the  use  of  those  instru- 
ments and  the  defects  to  which  they  were 
liable,  about  the  personal  error,  which 
was  an  important  point,  and  errors  from 
imperfect  adjustment.  The  conclusion 
at  which  he  had  arrived  with  regard  to 
continuous  integrators,  in  which  the  space 
passed  over  and  the  strain  were  multi- 
plied together  and  registered  continu- 
ously, was  that  they  were  exceedingly 
good  for  comparative  results,  but  were 
not  altogether  to  be  trusted  for  positive 
indications.  In  comparing,  for  example, 
a  number  of  machines  working  in  a  field 
under  similar  circumstances  as  to  weather 
and  everything  else,  with  the  same  oper- 
ator, the  comparative  results  would  be 
trustworthy  ;  but  if  there  were  any  vari- 
ation in  any  of  the  conditions,  they  would 
not.  There  was  always  a  good  deal  of 
doubt  about   the  positive  results.     The 


160 

causes  of  error  were  these.  In  the  con- 
tinuous integrators  the  integrating  wheel 
was  attached  to  a  train  of  wheel -work 
which  possessed  a  considerable  amount 
of  inertia  and  friction.  In  planimeters, 
and  integrators  of  that  class,  where  the 
observations  could  be  made  slowly,  at  a 
steady  speed,  and  where  the  conditions 
did  not  vary,  inertia  did  not  count  for 
much  ;  but  in  the  steam-engine,  and  in 
agricultural  implements  or  in  traction  in- 
dicators, there  were  great  and  rapid  vari- 
ations of  speed.  The  sudden  strains 
which  were  put  on  hj  the  tractive  force 
shifted  the  integrating  wheel  along  the 
disk  suddenly,  and  the  speed  changed  in 
a  similar  manner.  The  force  necessary 
to  accelerate  the  movement  of  the  inte- 
grating wheel  and  its  mechanism  tended 
to  cause  a  slip,  which  was  partly  counter- 
acted by  a  slip  produced  by  the  force 
necessary  to  arrest  its  motion  when  the 
speed  changed,  but  the  friction  of  the 
mechanism  always  acted  in  the  same  di- 
rection, tending  to  augment  the  error. 
"With  one  implement,  for  example,  if  there 


161 


were  a  tolerably  steady  pull  with  no  sud- 
den variation  in  the  velocity,  there  would 
be  one  amount  of  error,  whereas  if  an- 
other implement  were  worked  in  a  jerky 
fashion  there  would  be  a  totally  different 
amount  of  error.  With  regard  to  per- 
sonal error,  to  which  the  author  did  not 
attach  much  importance,  he  had  reason 
to  think  that  it  was  of  great  consequence. 
The  degree  of  care  and  skill  exhibited 
in  adjusting  the  instrument  and  taking 
the  measurements  had  an  important  in- 
fluence ;  he  was  therefore  always  careful, 
in  a  series  of  experiments  with  agricultu- 
ral implements,  to  have  the  same  ob- 
server throughout,  if  possible,  because 
the  results  aimed  at  were  rather  compar- 
ative than  positive.  Still,  he  was  bound 
to  say  that,  with  care  and  experience, 
satisfactory  positive  results  could  be  at- 
tained. The  author  did  not  appear  to  be 
aware  of  the  extensive  use  which  the 
Royal  Agricultural  Society  had  made  of 
the  integrator  which  had  always  been 
known  as  Morin's,  but  it  appeared  that 
honor  had  been  ascribed  where  it  was 


162 


not  strictly  due  ;  probably  the  reputation 
of  the  great  French  mechanic,  who  had 
done  so   much  to   introduce  it,  had  ob- 
scured the  claims  of  the  real  inventor. 
For  the  last  thirty-five  years  the  Royal 
Agricultural  Society  had  used  continuous 
integrators,  and  he   thought   that  there 
was  no  one  more  competent  to  speak  of 
their   action    than  Sir  Frederick  Bram- 
well,  who  had   himself  conducted  many 
of  the  experiments.  The  issues  had  often 
been  very  important,  involving  the  for- 
tunes of   manufacturers    of    agricultural 
implements,  which  had,  in  a  great  meas- 
ure,   hung   upon  the  indications  of  the 
dynamometers.     Most  of  the  apparatus 
used  by  the  Eoyal  Agricultural  Society 
had  been  designed  and  constructed   by 
the  late  Mr.  C.  E.  Amos,  M.  Inst.  C.  E., 
and  by  his  son  Mr.  J.  C.  Amos,  who  for 
many  years  filled  the  office  of  Consulting 
Engineers.     With  reference  to  continu- 
ous indicators  for  steam-engines,  he  had 
little  or  no   experience.      He  had  tried 
them,  but   the  results  had  not   been,  so 
far,  satisfactory.    One  of  the  chief  defects 


1 


163 


was  the  difficulty  of  keeping  the  little  in- 
tegratiug  wheel  perfectly  free  from  flats. 
It  was  not  easy  to  find  any  metal  per- 
fectly uniform  throughout ;  and  if  it  were 
not  uniform,  a  fiat  would  soon  form,  and 
then  all  the  results  would  be  utterly  un- 
trustworthy, because  the  wheel  tended  to 
hesitate  at  the  fiat  place.  Formerly  in- 
tegrating wheels  were  made  of  gun-metal, 
cast  with  great  care,  and  under  a  great 
deal  of  head.  Latterly  they  had  been 
made  of  steel,  and  a  better  result  had 
been  obtained.  But  even  when  the  wheel 
wore  uniformly,  if  the  width  of  the  sur- 
face in  contact  with  the  disk  varied,  then 
again  there  was  a  source  of  error,  because 
the  surface  of  the  integrating  wheel  lying 
upon  the  disk  became  greater,  and  then 
there  was  uncertainty  as  to  the  true  di- 
ameter of  the  periphery  of  the  disk  on 
which  the  integrating  wheel  was  Avorking. 
The  only  way  of  eliminating  these  errors 
was  by  repeated  testing  of  the  appar- 
atus. 

Mr.  C.  Vernon  Boys  said  the  first  part 
of  the  paper  on  which  he  desired  to  say 


164 

a  few  words  was  the  division  of  the  sub- 
ject into  different  classes.  The  author 
had  referred  to  one  system  of  classifica- 
tion which  Mr.  Boys  had  adopted  in  a 
jDaper  published  in  the  Philosophical 
Magazine,  a  division  into  three  classes. 
The  author  had  rightly  shown  that  the 
two  classes  which  Mr.  Boys  had  called 
the  "  radius  class,"  and  the  "  Amsler 
class.''  were  in  a  mechanical  sense,  that 
was,  so  far  as  the  connection  between 
the  surface  of  the  integrating  wheel  or 
roller  and  the  surface  on  which  it  worked 
was  concerned,  absolutely  identical.  But 
though  that  was  undoubtedly  the  case,  he 
thought  the  division  might  still  hold 
good,  for  an  inventor  could  not  have  con- 
trived machines  in  one  class  or  in  the 
other  without  having  had  some  such  sys- 
tem of  division  in  his  mind  at  the  time 
of  the  invention.  There  was  one  con- 
siderable omission  in  the  paper,  and  that 
was  the  only  point  on  which  he  felt  it 
necessary  to  find  fault.  There  was  no 
mention  of  a  very  large  series  of  most 
beautiful  machines,  designed,  and  he  be- 


165 


lieved  partly  constructed,  by  the  author 
himself.  As  those  mstruments  had  been 
fully  described  in  a  paper  by  Professor 
Shaw  before  the  Royal  Society,  possibly 
he  thought  that  they  were  so  well  known 
that  it  was  unnecessary  to  describe  them 
again ;  but  it  would  certainly  have  ren- 
dered the  paper  far  more  complete  and 
valuable  if  that  large  amount  of  work  had 
been  incorporated  in  it.  Of  all  the  in- 
struments brought  before  them,  he 
thought  that  the  new  precision  phmim- 
eters  and  that  extraordinary  spider- look- 
ing instrument  criuvling  on  the  sphere, 
were  those  which  called  for  the  utmost 
admiration.  It  was  impossible  to  look 
at  them  or  to  use  them  without  being 
impressed  with  the  extreme  mechanical 
beauty  of  their  construction  and  design. 
But  though  that  was  undoubtedly  the 
case,  he  thought  that  no  one  could  see 
some  of  the  combinations  of  a  sphere 
with  three,  eight,  or  six  little  w^heels 
round  it,  inventions  of  the  author,  with- 
out classing  them  even  above  those  in- 
struments in  point  of  beauty. 


166 

The  author  had  spoken  of  the  tangent 
principle  as  being  a  particular  case  of 
machines  which  worked  without  slipping. 
This  was  perfectly  true  ;  but  there  was  a 
very  great  distinction  between  integrators 
dej^ending  upon  the  tangent  action  and 
integrators  depending  upon  a  rolling  ac- 
tion of  the  radius  class,  to  which  alone 
the  other  machines  belonged.  There  was 
one  accidental  mistake  in  the  wall  dia- 
grams representing  two  spheres  which  he 
desired  to  point  out,  as  it  was  a  little 
puzzling.  In  the  instrument  of  Clerk 
Maxwell,  one  of  the  non-slipping  radius 
class,  the  axes  of  the  two  sp'ieres  were 
shown  in  such  a  position  that  rolling 
would  ultimately  make  them  coincide.  In 
reality,  the  equator  of  one  rolled  over  the 
pole  of  the  other,  as  was  obvious  to  those 
who  had  anything  to  do  with  integrators. 
The  distinction  between  instruments  of 
the  tangent  class  and  those  of  the  radius 
class  might  be  represented  by  the  little 
wheel  of  the  instrument.  The  accuracy 
of  integrators  of  the  radius  class  depended 
upon  the  exact  size  of  the  wheel,  and  the 


I 


167 

exact  size  of  the  surface  upon  which  it 
rolled;  and,  as  Mr.  Anderson  had  re- 
marked, it  was  very  much  impaired  when 
flats  formed  upon  any  of  the  surfaces  in 
contact,  for  there  was  then  a  little  hesita- 
tion. The  mere  action  of  integrating 
machines  in  which  there  was  slipping  was 
sure  to  produce  those  flats  at  some  time 
or  other,  so  that  the  time  they  were  likely 
to  last  and  the  amount  of  work  they  were 
capable  of  doing  were  limited.  The  actual 
size  of  the  roller  was  of  importance,  for 
as  it  wore  and  became  gradually  smaller, 
the  number  of  rotations  were  affected, 
and  therefore  the  recorded  result  gradu- 
ally increased.  These  were  the  objections 
to  instruments  of  what  he  had  called 
the  radius  type.  In  the  class  of  instru- 
ments at  which  he  had  worked  exclusively, 
and  which  he  believed  he  had  originated 
until  he  found  that  to  a  certain  extent 
Mr.  B.  Abdank- Abakan owicz  had  pre- 
ceded him,  the  wheel  was  allowed  to  roll 
along  just  in  the  direction  in  which  it 
was  pointing.  Anyone  who  had  been  in 
the  streets  of  London  must  have  noticed 


168 

with  wliat  extraordinary  persistence  and 
power  the  wheels  of  all  vehicles  went  in 
the  direction  in  which  they  were  point- 
ing. A  butcher's  cart  driven  round  a 
corner  with  tremendous  fury  would  often, 
in  sjDite  of  its  jumping  on  the  ground, 
still  continue  its  course  with  very  little 
side  slip.  If  the  springs  were  sufficiently 
good,  or  the  road  sufficiently  smooth, 
so  that  the  wheel  kept  in  contact  with 
the  ground  it  would  apparently  not  slip 
at  all.  In  the  case  of  the  first  instrument 
which  he  had  made,  which  he  had  called 
the  cart  integrator,  he  was  astonished  to 
find  with  what  extraordinary  accuracy 
results  were  obtained  by  its  use,  such,  for 
example,  as  the  area  of  a  circle  and  other 
things  that  were  known.  In  that  instru- 
ment he  had  depended  entirely  upon  the 
fact  that  the  steering  wheel  of  a  tricycle 
would  go  along  in  the  direction  in  which 
it  pointed.  The  instrument  had  to  pull 
a  heavy  brass  cart  after  it  and  slide  in  a 
comparatively  roughly-made  groove,  but 
even  so  he  found  the  value  of  ;r,  on  squar- 
ing the  circle,  came  out  3.14  when  using 


169 

a  very  rough  home-made  instrument.  It 
seemed,  then,  that  if  the  sources  of  side 
slip  which  were  undoubtedly  present, 
due  to  the  great  friction  that  had  to  be 
overcome  in  dragging  the  cart,  could  be 
removed — that  was,  if  the  ground  under 
the  cart  could  be  made  easily  movable, 
which  was  the  case  when  it  was  converted 
into  a  cylinder,  the  cylinder  would  follow 
exactl}^  in  the  direction  in  which  it  should 
go.  For  that  reason  an  integrator  of 
that  class  was  free  from  the  objections 
which  had  been  raised,  that  in  time  inte- 
grators wore  out,  the  little  wheel  got 
flats  upon  it,  and  as  the  size  varied  the 
record  varied.  In  the  case  in  point  there 
was  no  slii)ping  and  no  tendency  to  make 
flats,  and  if  the  wheel  was  made  half  or 
twice  the  size  the  record  was  the  same, 
for  it  only  depended  on  the  direction  in 
which  it  wanted  to  go,  not  upon  the 
amount  that  the  wheel  turned  in  going 
along  in  that  direction.  He  desired  to 
say  a  word  or  two  with  regard  to  an  in- 
strument, his  engine-power  meter,  which 
was  not  so  well  known  as  he  had  hoped 


170 

it  might  be.  There  were  certain  sources 
of  error  apparently  present  in  it,  but 
which  were  really  imaginary.  In  the 
first  place,  it  would  seem  that  if  the  posi- 
tion of  the  piston-rod,  and,  therefore,  the 
angular  position  of  the  little  tangent 
wheel  G  varied  in  the  least,  then  as  the 
cylinder  H  traveled  along  there  would  be 
an  error  due  to  the  angular  want  of  true 
precision.  But  that  was  not  so,  for  sup- 
posing the  spring  to  be  too  long  or  too 
short,  and  the  tangent  wheel  to  be  per- 
manently deflected,  when  there  was  no 
steam  pressure,  at  an  angle,  say,  of  2°  or 
3°,  then  as  the  cylinder  moved  in  one  di- 
rection, the  wheel  would  run  up  a  certain 
slope,  and  when  the  cylinder  moved  in  the 
other  direction,  it  would  run  back  along 
the  same  slope,  and  so  far  as  that  was 
concerned,  no  error  would  be  introduced. 
The  wheel  might  be  set  at  a  permanent 
angle,  and  the  roller  work  backwards  and 
forwards,  and  nothing  would  be  recorded 
at  the  end  of  the  operation.  He  could 
show  it  with  a  rough  paper  and  wood 
model.      If    he   permanently   compelled 


171 


the  tangent  wheel  to  assume  a  certain 
angle  by  holdmg  it  with  his  thumb,  as 
the  cylinder  traveled  in  one  direction,  it 
would  rotate  a  certain  amount,  and  when 
it  traveled  in  the  other  direction  it  would 
rotate  back  again,  and  on  the  dial  there 
would  be  no  permanent  record.  The 
disk-cylinder  integrator  had  one  serious 
defect ;  it  was  very  difficult  to  apply  it  to 
cases  in  which  growth  of  time  or  of  mo- 
tion was  continuous.  If  it  was  desired 
to  have  a  time-integral  and  a  motion-in- 
tegral, when  the  motion  was  continuous, 
it  could  not  be  easily  applied,  because 
when  the  cylinder  had  got  to  one  end  of 
the  stroke,  there  was  nothing  for  it  but 
to  stop  and  come  back  again.  It  was, 
therefore,  necessary  to  apply  a  mangle 
motion,  so  that  as  the  motion  was  con- 
tinuous the  cylinder  went  backwards  and 
forwards ;  then  by  causing  the  cylinder 
to  work  between  two  tangent  wheels  al- 
ternately, or  by  letting  the  mangle-motion 
work  ordinary  reversing  gear  between 
the  cylinder  and  the  recording  mechan- 
ism, continuous  integration  could  be  ef- 


172 

fected.  But  of  course  the  instrument, 
where  continuous  integration  was  con- 
cerned, would  not  compare  with  the 
spherical  integrators  designed  by  the 
author.  On  the  other  hand  he  did  not 
think  that  any  instrument,  in  the  peculiar 
case  presented  by  the  automatic  integra- 
tion of  an  engine  diagram,  could  compare 
with  it,  for  those  peculiarities  of  motion 
which  interfered  with  the  ordinary  form 
of  radius  machines,  by  causing  a  perpet- 
ual scrubbing,  to  which  reference  had 
been  made,  produced  no  trouble  at  all, 
for  there  was  no  side-slipping,  and  be- 
cause the  very  large  moment  of  the  radius 
integrator  was  replaced  by  the  extremely 
small  moment  of  a  little  cup  containing 
a  wheel  not  the  size  of  a  three-penny  bit, 
and  by  a  little  bar  of  steel.  In  fact,  the 
moment  of  the  piston  and  its  attachments 
was  nothing  like  so  great  as  that  of  an 
ordinary  Richards  Indicator,  because 
there  was  no  multii^lication  of  motion. 

Mr.  W.  Anderson  wished  to  state  that 
he  had  been  able  to  bring  to  the  meeting 
the  integrating  part  of  one  of  the  old 


173 

dynamometers  of  the  Royal  Agricultural 
Society.  The  date  of  its  construction 
was  unknown,  but  he  believed  it  w^as 
about  the  year  1848. 

■  Mr.  J.  G.  Mair  observed  that  the  au- 
thor had  not  given  himself  sujB&cient 
credit  for  the  machine  he  had  invented, 
and  he  would  like  to  ask  him  if  the  read- 
ing of  the  counter  gave  such  accurate 
results  as  the  vernier  on  Amsler's  plan- 
imeter.  To  read  by  means  of  a  counter 
was  of  great  advantage,  and  especially  so 
where  large  numbers  of  indicator  dia- 
grams had  to  be  taken  out ;  on  some  of 
the  engine  trials  he  had  made,  three 
hundred  diagrams  had  to  be  averaged,  and 
constantly  reading  the  vernier  on  Profes- 
sor Amsler's  instrument  was  trying  to 
the  eyes.  The  application  of  the  inte- 
grating machine  as  a  power  meter  was  a 
most  useful  one,  and  he  had  made  several 
trials  with  the  one  invented  by  Mr.  0. 
Vernon  Boys ;  on  three  trials  the  read- 
ings were  68.5,  68.2,  and  73  by  the  me- 
ter, against  67,  67.5,  and  72  in  the  pump. 
The  pump-power  was  taken  from  the  dis- 


174 

placement  of  the  pump  piston,  and  the 
head  as  shown  on  a  mercury  gauge.  He 
thought  those  results  were  a  proof  of  the 
correctness  of  the  iustrumeut.  He  did 
not  think  that  reading  a  counter  was 
more  difficult  than  reading  the  dial  of  a 
watch.  One  other  measurement  had  to 
be  made,  namely,  the  distance  between 
the  stops,  and  as  that  could  be  measured 
with  an  ordinary  rule,  a  child  could 
almost  make  the  simple  calculation,  so 
that  he  did  not  think  personal  error  very 
much  affected  such  an  instrument.  There 
was  naturally  a  good  deal  of  diffidence  in 
using  a  machine,  the  details  of  which 
were  not  thoroughly  grasped.  With  an 
indicator  diagram,  there  was  something 
to  see  which  was  readily  understood ;  but 
where  compound  measurement  was  shown 
on  a  counter,  it  was  at  first  difficult  to 
realize  the  reading  as  correct.  As  soon, 
however,  as  the  instruments  were  better 
known,  he  had  no  hesitation  in  saying 
that,  where  the  absolute  shape  of  an  in- 
dicator diagram  was  of  no  consequence, 
the  continuous  integrator  would  entirely 


175 

supersede  all  other  means  of  measuring 
power. 

Mr.  Druitt  Halpin  said  that  the  author 
had  referred  to  one  of  the  minor  im- 
provements by  putting  a  locking-spring 
to  the  frame.  Mr.  Mair  had  noticed  the 
great  difficulty  there  was  in  reading  the 
instrument,  on  account  of  the  small  scale 
to  which  it  was  graduated,  l)ut  the  author 
had  not  completely  followed  the  idea  of 
adding  the  locking  arrangement.  The 
convenience  of  reading  the  instrument 
was  doubtless  very  great,  for  it  could  be 
taken  up  and  put  in  a  good  light,  but  the 
real  object  of  the  locking  gear  was  to 
make  the  instrument  do  its  own  addition, 
and  also  carry  over  the  decimal  places 
now  lost.  Instead  of  taking  each  dia- 
gram by  itself,  and  writing  down  the 
result  of  each  separately,  and  adding 
them  up,  the  instrument  w^as  set  at  0  and 
and  locked  ;  the  instrument  was  run  over 
the  diagram,  and  it  was  locked,  and  so  on. 
So  that  by  taking  ten  diagrams  succes- 
sively, and  putting  the  decimal  point  one 
point  back,  the  instrument  was  made  to  do 


176 


its  own  addition  It  bad  the  further  ad- 
vantage that,  whatever  the  last  decimal 
might  be,  it  was  carried  on  by  the  instru- 
ment. If  it  was  2.38,  it  could  not  be 
said  whether  it  w^as  2.389  or  2.381.  With 
regard  to  Mr.  Boys'  application  of  these 
instruments,  he  had  had  an  opportunity 
of  testing  it  on  an  engine  of  120  or  130 
I.  H.  P.,  and  he  found  the  results  coin- 
cided within  from  1  to  2  per  cent,  of  the 
indicated  power  which  was  obtained  with 
standard  indicators.  With  Mr.  Webb's 
permission,  he  also  put  one  of  his  power- 
meters  on  the  large  rail  mill  at  Crewe, 
and  took  a  series  of  observations  there 
of  the  exact  power  required  to  roll  a  rail, 
from  the  moment  the  ingot  touched  the 
rolls  to  the  moment  the  next  ingot 
touched  them,  which  gave  the  true  pow- 
er, correction  being  made  for  any  differ- 
ence of  velocity  in  the  fly-wheel.  It  gave 
the  power  both  while  the  engine  was 
running  with  the  bloom  in  the  mill  and 
when  it  was  running  empty.  Whether 
the  machine  in  its  j)resent  form  would  be 
suitable  for  locomotives,  he  was  hardly 


177 

prepared  to  say,  because  he  feared  that 
the  attachment  which  was  provided  for 
taking  diagrams  was,  perhaps,  so  heavy 
that  its  inertia  would  interfere  with  the 
correct  action  of  the  instrument.  But,  if 
that  was  Hghtened,  or  the  attachment 
left  out,  he  was  sure  the  best  results 
would  be  obtained  from  it. 

Mr.  H.  Cunynghame  said  that  his  at- 
tention had  been  called  a  great  deal  to 
Mr.  Boys'  machines,  in  the  development 
of  which  he  had  assisted,  as  they  were 
first  designed  for  practical  application  to 
steam-engines.  He  had  made  many  at- 
tempts to  apply  continuous  integrators 
for  the  purpose  of  integrating  electric 
power,  and  he  certainly  could  safely  bear 
out  all  that  Mr.  Anderson  had  said  about 
the  imperfections  of  integrators  of  the 
slipping  type.  If  a  wheel  was  running 
in  any  direction  in  which  it  was  likely  to 
slip  upon  the  surface,  it  would  be  rubbed 
into  facets,  and  in  like  manner,  a  sphere 
would  be  rubbed  into  a  polyhedron.  In 
this  state  both  wheel  and  sphere  would 
be  worse   than  useless  for   integration. 


178 


But  if  these  machines  had  an  integrator 
of  the  roller  type,  even  if  it  had  facets 
already,  the  rolling  wonld  take  the  facets 
out  of  it  just  as  an  apothecary  rolled  his 
pills.  Instead  of  an  integrating  wheel 
of  hard  steel,  as  was  necessary  in  all  in- 
tegrating machines  of  the  slipping  type, 
in  those  of  the  rolling  type,  soft  metal 
might  be  used,  and  the  more  it  was  used 
the  better  it  would  get.  Moreover,  in 
such  machines,  since  what  was  counted 
was  not  the  revolutions  of  the  wheel,  but 
the  revolutions  of  something  caused  to 
roll  by  means  of  the  wheel,  the  accuracy 
of  shape  of  the  rolling  wheel  became  of 
minor  importance,  and  even  if  it  " skated" 
over  the  surface  instead  of  rolling,  a  fair 
result  would  be  obtained.  He  thought 
Mr.  Anderson's  remarks  were  somewhat 
unjust  to  planimeters  of  the  rolling  type, 
which  he  did  not  think  had  been  brought 
before  the  public  so  as  to  be  within  the 
experience  of  anyone.  That  led  him  to 
say  a  few  words  on  Mr.  Boys'  steam- 
power  meter.  He  believed  that  Mr.  Boys 
and  he  were  the  first  in  this  country  to 


179 

make  a  trial  of  those  machines.  They 
made  a  trial  at  the  works  of  Messrs.  Ran- 
some  &  Jocelyn  at  Battersea,  and  for 
that  purpose  they  had  an  extremely  good 
indicator  made  by  Messrs.  Elliot.  It 
would  be  quite  understood  that  Mr. 
Boys'  indicator  was  not  in  any  way  cali- 
brated by  mere  trial.  They  took  the 
measurements  of  the  engine  and  the  di- 
ameter of  the  cylinder.  Then  thej^  took 
the  amount  by  which  a  certain  pressure 
of  steam  would  cause  the  spring  to  rise, 
just  as  was  done  before  taking  the  H.  P., 
by  means  of  a  Richards  Indicator.  They 
thought  that  if  those  two  machines  were 
used  upon  the  same  engine,  and  if  the  re- 
sults given  by  the  two  machines,  namely, 
Richards  and  Boys,  were  independently 
calculated  from  the  constants  of  the  two 
machines,  then,  if  these  results  nearly 
corresponded,  a  very  remarkable  coinci- 
dence of  testimony  would  be  obtained. 
They  therefore  put  Boys'  machine  on  the 
engine  alternately  with  Richards',  and  he 
thought  they  must  have  alternated  twenty 
times,  and  the  results  of  Boys'  machine 


180 

were  found  to  be  uniformly,  during  the 
first  series  of  experiments,  about  23  per 
cent,  too  high.  That  puzzled  them  ex- 
tremely, but  upon  examination  they  found 
that  the  manufacturer,  instead  of  making 
the  piston  1  inch  in  diameter,  had  made 
it  1  square  inch  in  area,  and  when  they 
made  the  proper  correction  they  found 
the  results  to  agree  to  about  Ih  per  cent., 
and  with  those  stated  by  Mr.  Hal^Din  and 
Mr.  Mair.  Mr.  Mair's  tests  were  made 
in  this  way:  he  raised,  by  means  of  a 
pump,  a  given  weight  of  water  through  a 
number  of  feet,  and  then  he  estimated 
the  foot  pounds  and  compared  them  with 
one  another,  and  the  result  showed  a  re- 
markable degree  of  accuracy,  the  discrep- 
ancy of  one  or  two  per  cent,  being  account- 
ed for  by  the  loss  of  work  owing  to  the 
raising  of  the  pump  valves,  which  were 
large  and  heavy. 

Dr.  William  Pole,  after  testifying  to 
the  high  character  of  the  paper,  offered 
a  few  remarks  on  the  very  early  example 
of  mechanical  integration  with  which  the 
author  had  connected  his  name.     It  had 


181 


come  about  in  the  following  manner : 
Some  half  century  ago  the  engineers  of 
the  center  and  north  of  England  became 
aware  of  the  reports  published  from 
time  to  time  of  the  extraordinary  econ- 
omy of  the  pumping  engines  of  the  mines 
of  Cornwall.  These  reports  at  first  ob- 
tained no  credence,  and  even  when  they 
were  found  to  have  some  foundation,  the 
most  singular  attempts  were  made  to 
explain  them  away.  In  the  midst  of  the 
controversy,  the  late  Mr.  Thomas  Wick- 
steed,  M.  Inst.  C.  E.,  the  Engineer  to  the 
East  London  Water  Works  Company, 
determined  to  throw  light  on  the  ques- 
tion by  buying  an  engine  in  Cornwall, 
and  setting  it  up  to  pump  water  on  his 
own  premises  at  Old  Ford,  where  it 
could  be  thoroughly  tested  and  exam- 
ined. 

The  subject  had  previouslj^  been 
brought  before  the  notice  of  the  British 
Association  for  the  Advancement  of  Sci- 
ence, and  had  attracted  the  attention  of 
Professor  Henry  Moseley.  He,  in  writ- 
insr  his    excellent   work    on  "  The    Me- 


182 

chanical  Principles  of  Engineering  and 
Architecture,"  had  become  acquainted 
with  a  principle  of  dynamometrical  ad- 
measurement proposed  by  Mr.  Ponce- 
lot,  and  carried  out  in  1883  by  General 
Morin  ;  and  it  occurred  to  him  that  a 
machine  might  be  contrived  on  a  similar 
principle,  applied  to  record  the  work 
done  by  a  steam-engine,  l)y  a  species  of 
mechanical  integration,  combining  the 
pressure  exerted  on  the  piston  with  the 
space  moved  through  ;  and  it  was  seen 
that  such  a  machine  would  be  most  use- 
fully applied  in  testing  the  performance 
of  Mr.  VVicksteed's  Cornish  engine. 

At  the  meeting  of  the  Britisli  Associa- 
tion in  ]  840,  a  grant  was  made  for  the 
purpose,  and  a  committee,  consisting  of 
Professor  Moseley,  Mr.  Eaton  Hodgkin- 
son,  and  Mr.  J.  Enys,  was  a2)pointed  to 
carry  it  out.  The  machine  was  con- 
structed under  Professor  Moseley's  di- 
rection, and  a  full  account  of  it  was  given 
in  the  report  for  the  following  year. 
The  trials  were  then  made  on  the  engine, 
and   the  results  were    exceedingly  satis- 


183 


factory.  The  integrator  worked  for  a 
month  without  intermission,  and  its  in- 
dications, when  calculated  out,  were 
found  to  agree  closely  with  the  results 
obtained,  as  accurately  as  they  could  be, 
by  other  means. 

The  fixed  data  of  the  engine  had  all 
been  well  ascertained,  but,  to  render  the 
comparison  complete,  it  was  found  de- 
sirable to  get,  if  possible,  an  accurate 
measurement  of  the  velocity  of  the  pis- 
ton at  various  parts  of  its  stroke  ;  this 
velocity  was  very  variable,  depending  not 
only  on  the  mass  in  motion,  but  also  on 
the  ever-varying  force  of  steam  acting 
on  the  piston,  and  on  means  which  had 
hitherto  been  devised  for  ascertaining 
the  velocity  experimentally.  The  atten- 
tion of  the  committee  had,  however, 
been  directed  to  an  admirable  chrono- 
metrical  instrument  contrived  by  Messrs. 
Poncelot  and  Morin,  and  Professor 
Moseley  undertook  to  adapt  a  machine 
on  this  principle  to  the  Old  Ford  en- 
gine. 

At  this  time  Dr.  Pole  (who  had  been 


IS  I 


oooupioil  iuiioptMultutlv  it\  invostijjfrtting 
tho  rtoiion  of  the  OiU'uish  eiij^ino)  \\i\i\ 
the  honor  of  bein*j'  itwittnl  to  join  the 
eoiniuittee»  ivnd  the  two  siu'cotnlino-  re- 
ports, \\\  IS4;>  ixud  \SAA,  wore  writttMi  hv 
hiiu. 

By  luertus  of  these  two  iustnimeiits, 
and  by  t\\\  ordinary  indieator,  aided  by 
eai^eful  observiitions  ns  to  the  consump- 
tion of  ooa\  i\\\<\  water,  and  other  vari- 
able factors,  the  workin*^  of  the  OKI  Ford 
Cornish  engine  was  investigated,  both 
theoretieally  and  practically,  i!\  a  most 
con^plete  and  accunUe  maniier,  and  the 
peculiarities  of  this  fornt  of  engine,  as 
comjvared  with  the  oi\linary  engines  iu 
use  at  that  time,  weit)  thoi\^ughly 
brought  out,  so  a^  to  clear  up  all  the 
doubts  that  had  been  so  long  entertained 
by  eugineei*s.  This  application  of  a  me- 
chanical integnUor  to  i-eal  work  of  mag- 
nitude and  importance  was  probably  the 
earliest  made.  It  was  intendevl  to  fol- 
low it  \ip  by  adapting  the  machine  to 
other  purposes,  especially  to  oi»ean-goiug 
steamers,  and  it   was    actually   fitted   to 


the  cTjf^iueH  of  the  (Ireat  Wehtern  htearu- 
ship  un  her  firnt  voyaj^e,  but  by  an  acci- 
dent which  it  waH  difficult  to  repair  at 
sea,  the  experiment  wan  rendered  UKe- 
lesB,  and  the  attempt  wan  never  re- 
newed. 

Mr,  \V.  li.  iJouHlield  naid  ho  liad  nut 
had  the  advantage  of  reading  the  proof 
of  ProfeHBor  Shaw'n  paper,  but  Jjc  had 
heard  it  read  in  ith  nhort  form  at  the 
previous  meeting,  and  he  thought  po»- 
ftibly  it  miglit  be  of  inferent,  and  uneful 
in  connection  with  a  paper  whicij  dealt, 
more  or  lesB,  with  the  different  typen  of 
mer.'hitnical  integratorn,  if  he  mentione<l 
a  principle  which,  bo  far  as  he  knew,  had 
not  been  alluded  to.  It  occurred  to  him 
8ome  years  ago  that,  Vy  the  use  of  vari- 
ous curves  in  connection  with  planim- 
eters,  various  products  and  squares 
might  be  reaxlily  integrated.  A  simple 
form  of  area  plani meter  of  this  kind 
might  be  made  Vjy  combining  a  T-square 
and  a  set-square,  or  two  set-squares. 
Referring  to  the  illustration^  Fig.  57,  the 
lower  T-square  or  set-sqtiare  ABC  would 


186 


have  in  it  a  slot  KL,  and  upon  it  a  rib 
GH,  and  would  be  capable  of  motion 
about  a  point  R  in  line  with  the  slot. 
The  upper  set-square  would  have  a  slot 
MN  capable  of  sliding  upon  the  rib  GK, 
and  also  a  parabolic  groove  OP.  A 
pointer — in  practice  a  ring  with  cross 
wires— would  slide  in  the  slots  KL  and 
OP.  In  using  this  planimeter  a  needle 
point  would  serve  to  pin  the  lower  set- 
square  through  the  point  R,  at  a  conve- 
nient pole  in  relation  to  the  area  to  be 
integrated.  The  cross-wires  or  pointer 
would  be  carried  round  the  curve  enclos- 
ing the  area  to  be  integrated,  the  two 
set-squares  at  the  same  time  sliding 
on  one  another  by  means  of  the  rib 
GH  and  slot  MN.  The  measurement 
would  be  made  by  the  roller  Q  placed 
at  the  vertex  of  the  parabolic  curve. 
The  principle  of  the  apparatus  was  very 
simple.  Taking  polar  co-ordinates,  with 
R  as  the  pole,  the  area  to  be  integrated 
could  be  expressed  as 


i 


187 


Fiir.  oT 


-Kv.  - 


AREA  PLANIMETER 


188 

If  now  the  equation  of  the  parabolic 
slot  were 

and  the  parabola  moved,  as  in  the  appar- 
atus, so  that  y  was  always  =r,  the  ex- 
pression for  the  area  would  become 

f2axd  e. 

This  showed  that  the  area  was  meas- 
ured by  the  roller  Q  at  the  vertex  of  the 
parabola.  This  proof  rather  suggested 
that  possibly  the  use  of  polar  co  ordi- 
nates  in  some  of  Professor  Shaw's  proofs 
would  shorten  them.  By  the  use  of 
other  curves,  instead  of  a  parabola,  the 
principle  could  be  applied  to  the  me- 
chanical solution  of  various  problems  in 
integration. 

Professor  W.  H.  H.  Hudson  remarked 
that  the  classification  adopted  by  the  au- 
thor coincided,  if  not  entirely,  very  con- 
siderably with  this,  that  in  the  instru- 
ments in  which  slipping  was  bound  to 
take  place,  when  it  was  attempted  to  ex- 
press the  element  of  area,  in  mathemati- 
cal language  y  dz,  dz  corresponded  to  the 


189 

angle  turned  through  and  y  to  the  radius, 
so  that  the  area  corresponded  to  the 
length.  That  was  certainly  so  in  some 
of  the  earlier  instruments  ;  but  those  in 
which  no  slipping  w^as  allowed  to  take 
place  seemed  to  correspond  rather  to  a 
different  mode  of  translating  the  mathe- 
matical expression.  There  was  a  certain 
amount  of  turning  which  was  propor- 
tionate to  dz^  and  then  by  a  suitable 
mechanism  that  was  turned  into  some- 
thing else  proportional  to  y  dz  by  using 
a  multiplier  proportional  to  y.  One 
wheel  was  made  to  turn  so  many  times 
faster  than  another,  the  number  of  times 
corresponded  to  y  instead  of,  as  in  the 
first  class  of  instruments,  the  radius  be- 
ing proportional  to  y.  That  multiplier 
occurred  in  the  various  forms  of  the  in- 
strument, sometimes  as  a  sine,  and  some- 
times as  a  tangent.  Those  of  the  sine 
class  w^ould  have  this  disadvantage,  that 
the  multiplier  must,  of  necessity,  be  a 
proper  fraction.  That  would  limit  the 
range,  whereas  those  dependent  on  the 
tansfent  seemed   to    have    no    limitation 


190 


whatever,  and  the  multiplier  might  be 
anything.  This  seemed  to  be  a  great  ad- 
vantage of  the  sphere  instrument  of  Pro- 
fessor Shaw,  that  there  was,  apparently, 
no  limitation  at  all,  that  which  represent- 
ed y  being  a  multiplier  which  represent- 
ed a  tangent.  He  had  always  taken 
great  interest  in  mechanical  illustrations 
of  mathematics.  In  most  cnses  the 
mathematics  had  been  incidental,  but  in 
this  case  it  was  of  the  very  essence  of 
the  whole  business.  These  were  instru- 
ments designed  to  illustrate  not  only 
mathematical  results,  but  even  mathe- 
matical processes,  and  he  was  very  much 
struck  by  the  way  in  which  the  author 
had  developed  Amsler's  planimeter  out 
of  the  simple  form  of  the  disk  and  roller. 
It  illustrated  a  mathematical  conception 
which  was  sometimes  received  with  in- 
credulity, by  those  who  met  with  it  for 
the  first  time,  namely,  treating  parallel 
straight  lines  as  meeting  in  a  point  at  in- 
finity. This  was  done  in  pure  mathe- 
matics, and  he  found  that  the  author  had 
been  obliged  to  do  it  in   order  to   bring 


I 


191 


his  instruments  all  under  one  head.  It 
seemed  to  him  that  this  was  a  striking 
illustration  of  the  inter-action  of  mathe- 
matics and  mechanics,  and  the  benefit 
which  each  gave  to  the  other. 

Mr.  W.  E.  Rich  said  the  integra- 
tion of  the  areas  of  plane  surfaces,  which 
these  instruments  were  ordinarily  de- 
signed to  do,  was  much  less  difficult  than 
the  integration  of  work  done  on  a  dyna- 
mometer in  the  manner  spoken  of  by 
^Ir.  Andersnn.  One  of  the  great  desid- 
erata in  dynamometrical  measurements 
was  to  get  a  spring  which  followed 
Hooke's  law  ;  but,  unfortunately,  in  prac- 
tice they  never  got  one  that  followed  it 
precisely.  The  uext  thing  was  to  adjust 
the  instrument  for  zero.  If  the  instru- 
ment were  not  accurately  so  adjusted, 
complex  formulas  had  to  be  used  for  re- 
ducing the  results,  and  as  in  many  in- 
stances zero  was  not  accurately  deter- 
mined, and  these  formulas  were  at  the 
same  time  overlooked,  the  results  record- 
ed were  in  such  cases  very  erroneous. 
One  point  of  practical  difficulty  with  in- 


192 

tegration  for  a  dynamometer  ^was  the 
slipping  of  the  small  disk  upon  the 
large  one.  In  the  field,  when  it  was 
raining,  the  observers  were  much  ham- 
pered in  that  way,  and  on  a  stony  road, 
when  traveling  at  a  high  speed  with  a 
small  integrating  disk,  the  contact  be- 
tween the  disks  was  frequently  broken  by 
the  shake  and  jar  of  the  whole  instru- 
ment. These  observations  were  perhaps 
not  directly  pertinent  to  the  question  of 
integration ;  but  one  of  the  most  useful 
objects  of  integration  was  for  dynamomet- 
rical  purposes,  and  therefore  he  thought 
it  worth  while  to  mention  them. 

Professor  H.  S.  Hele  Sbaw,  in  reply, 
said  that  the  question  of  mechanical  in- 
tegrators was  a  very  much  wider  one 
than  many  would  at  first  think ;  but  the 
discussion  upon  the  paper  had  embraced 
two  principal  points:  Planimeters  such 
as  were  shown  upon  the  table,  and  the 
subject  of  Continuous  Integrators.  The 
shpping  class  of  plaiiimeter  bad  been 
brought  to  a  great  state  of  perfection, 
and  their  accuracy  was,  as  he  had  shown. 


193 

remarkable.  It  seemed  to  him  that  the 
only  objection  to  them — an  objection 
which  several  speakers  had  adverted  to 
— was  the  diflSculty  of  reading  them. 
Most  of  the  remarks  had  been  devoted 
to  continuous  integrators.  Perhaps  that 
was  not  altogether  unnatural,  because 
they  had,  as  Mr.  Rich  had  pointed  out, 
such  a  directly  practical  application. 
Mr.  Anderson  had  given  valuable  testi- 
mony on  the  subject  of  the  defects  of 
such  instruments  as  were  in  use  for  dy- 
namometrical  purposes,  and  his  experi- 
ence agreed  with  that  of  Mr.  Rich  and  of 
all  who  had  much  occasion  to  use  them. 
Professor  Shaw  thought  the  general  con- 
clusion was  that  none  of  the  continuous 
integrators  in  use  for  this  purpose  were 
really  satisfactory  in  their  action.  To 
say  of  an  instrument  of  measurement 
that  it  was  not  an  accurate  instrument, 
that  it  was  merely  capable  of  giving  com- 
parative results,  was.  after  all,  to  give  it 
very  poor  praise,  for  mere  comparative 
results  simply  meant  that  no  constant 
for  the    instrument    could    be   obtained. 


194 

He  bad  no  doubt  that,  for  temporary 
purposes,  the  difficulties  might  be  over- 
come, but  it  was  the  general  opinion 
that  an  instrument  like  the  disk  and 
roller,  which,  by  the  way,  in  spite  of 
what  Mr.  Anderson  had  said,  ica?^  first 
applied  by  Morin  after  its  invention  by 
Poncelot  before  1840,  was  not  reliable 
when  in  continuous  use.  An  instrument 
that  worked  one  day  differently  from  an- 
other day,  because  the  conditions  were 
slightly  changed,  must  certainly  be  pro- 
nounced to  be  very  imperfect.  It  was 
because  of  this  very  difficulty  that  he 
was  led,  as  he  had  no  doubt  Mr.  Boys 
and  others  had  been,  to  work  at  the  sub- 
ject, and  to  endeavor  to  overcome  the 
difficulty  of  the  slipping  action.  Mr. 
Mair,  Mr.  Druitt  Halpin,  and  Mr.  Cun- 
ynghame  had  all  testified  to  the  satisfac- 
tory action  of  the  tangent  integrator  of 
Mr.  Boys,  which  was  one  solution  of  the 
problem  of  avoiding  the  slipping  action 
in  question,  but  had  been  employed  in  a 
far  more  trying  apphcation  than  with  a 
dynamometer,  viz.,  a  steam-engine  integ- 


lt)5 


rator.  This  he  was  very  glad  to  hear, 
because  the  instruments  of  his  own,  ex- 
hibited on  the  table,  depended  upon  the 
same  principle  for  their  successful  action, 
and  it  was  just  in  this  particular  that 
they  differed  from  the  integrator  referred 
to  by  Dr.  Pole.  His  instruments  had 
been  alluded  to  in  complimentary  terms 
by  one  or  two  of  the  speakers,  and  per- 
haps he  might  be  allowed  briefly  to  de- 
scribe the  principle  upon  whicli  they 
worked,  and  also  to  call  attention  to  two 
new  instruments  wliich  were  sliown  for 
the  tirst  time  on  the  previous  Thursday 
at  the  Royal  Society.  The  difficulty 
which  he  endeavored  to  overcome  was  to 
make  use  of  those  mathematical  proper- 
ties of  the  sphere,  which  he  had  j^revi- 
ously  discovered  and  pubhshed,  and 
which  Professor  Hudson  had  alhided  to 
as  to  sine  and  tangent  forms.  As  was 
not  surprising,  he  hsid  found,  when  he 
came  to  collect  information  on  the  sub- 
ject, that  the  same  properties  had  been 
employed  by  others,  amongst  whom  were 
Professor  Mitchels(m  and  Professor  Ams- 


196 

ler.  But  what  the  author  had  endeav- 
ored to  do  was  to  avoid  the  sHpping  of 
the  measurmg  roller  over  the  integrating 
sphere.  That  was  his  first  thought. 
There  was  generally  a  germ  in  all  inven- 
tions, and  that  was  the  germ  of  his.  If, 
instead  of  moving  the  measuring  rollers 
round  the  sphere,  they  could  keep  them 
in  contact  with  the  sphere,  and  move  the 
axis  of  the  sphere,  or,  as  it  at  first  oc- 
curred to  him,  have  the  two  rollers  in 
contact  and  roll  them  around  the  spheres, 
they  would  hold  it  in  such  a  way  that 
they  would  allow  it  to  rotate  around  the 
different  axes  in  the  solid.  At  first  he 
did  not  think  it  would  work,  but  he 
tried  it  with  a  rough  boxwood  ball,  and 
two  disks  with  india-rubber  rings  arouDd 
them,  and  he  was  surj^rised  to  see  that 
the  rollers  did  move  round,  and  to  see 
the  ball  rolling  round  on  a  different  axis. 
He  thought  at  first  it  was  merely  a  ques- 
tion of  rolling  centers,  but  after  some 
time  he  saw  that  it  was  really  a  question 
of  geometrical  i^rinciple,  and  having  cer- 
tain rollers  in  contact  around  one   equa- 


197 

tor  it  did  not  matter  where  they  were- 
placed,  so  long  as  the  others  were  per- 
pendicular to  the  first,  acting,  in  fact,  on 
the  same  principle  as  that  by  which  two 
beveled  wheels  would  work  together- 
The  axis  of  rotation  around  which  the 
sphere  must  turn,  because  of  the  con- 
tact of  the  other  set  of  rollers,  was  in 
the  vertical  plane  ;  there  was  but  one  in- 
tersection of  the  two  plates,  and  that  in- 
tersection was  the  axis  of  rotation  of  the 
sphere.  That  principle  once  understood 
enabled  him  to  construct  various  other 
models  based  upon  it.  He  was  indebt- 
ed almost  entirely  to  his  brother,  Mr. 
Edward  Shaw,  Stud.  Inst.  C.  E.,  and  to 
Mr.  W.  E.  Kerslake,  for  working  out  this 
in  a  practical  form,  and  constructing  the 
various  instruments  exhibited.  One  of 
these  instruments  was  a  continuous  integ- 
rator, which 'had  been  designed  for  ap- 
plication to  the  dynamometer  of  the 
Rev.  F.  J.  Smith,  of  Taunton,  who  in- 
tended to  have  been  present  and  take 
part  in  the  discussion,  but  from  whom  he 
had  just  received  a  telegram  stating  that 


198 


he  had  tried  it  most  carefully  and  ex- 
haustively, and  bad  found  that  it  had 
given  absolutely  uniform  results.  He 
had  tried  it  himself,  but  it  was  a  very 
satisfactory  thing  that  it  went  to  Mr. 
Smith  straight  from  the  hands  of  the 
maker,  not  adjusted  in  any  way.  It  was 
remarkably  simple.  According  as  the 
arm  was  moved  into  a  more  or  less  in- 
clined position,  so  there  resulted  a  larger 
or  smaller  travel  of  the  indicator.  No 
appreciable  effort  was  required  to  turn  it ; 
it  was  done  with  a  minimum  of  power, 
and  it  entirely  got  over  the  difficulty  of 
unequal  wearing.  He  had  every  reason 
for  thinking  that,  if  anything,  it  would 
rather  tend  to  wear  the  ball  round  than 
otherwise.  Another  instrument  that  he 
had  on  the  table  was  the  moment  integ- 
rator. Professor  Amsler's  moment  in- 
tegrator was  an  expensive  one,  and  at 
the  time  of  writing  the  paper  he  had  not 
any  idea  of  constructing  one  :  but  di 
rectly  he  had  made  a  small  area  planim- 
eter,  he  saw  that  a  much  simpler  instru- 
ment could  be  devised  by  employing  the 


199 

wheels  in  a  very  small  compass,  and  he 
should  be  haj^py  to  explain  after  the 
meeting,  to  anyone  who  might  be  inter- 
ested, the  performance  of  the  operation 
by  the  application  of  these  small  wheels. 
Mr.  Mair,  he  believed,  asked  whether  the 
readings  of  the  instrument  were  accu- 
rate. It  had  not  been  tried  so  exhaust- 
ively as  Amsler's  instruments,  and  there- 
fore he  could  not  compare  it  with  them ; 
but  he  saw  no  reason  why  they  should 
not  expect  the  same  accuracy  from  it  as 
from  those  instruments  in  the  non-slip- 
ping class,  concerning  the  accuracy  of 
which  they  had  already  heard  testimony. 
He  greatly  regretted  that  it  had  not 
been  possible  to  prepare  the  engravings 
in  time  to  be  sent  around  with  the  paper, 
or  some  points  he  had  been  anxious  to 
have  discussed  would,  no  doubt,  have 
been  dealt  with  more  fully  by  the  vari- 
ous speakers.  For  instance,  the  points 
which  Professor  Hudson  had  referred  to 
were  actually  treated  in  the  paper,  and, 
he  ventured  to  think,  were,  by  means  of 
diagrams,    made    perfectly    clear.      He 


200 

would  remark  that,  with  reference  to  the 
range  of  multiplier  alluded  to,  although 
it  was  quite  true  that  this  was  much 
greater  in  the  case  of  the  tangent  form 
of  integrator,  in  fact  was  infinitely  great, 
yet  for  all  practical  purposes  the  sine 
form  enabled  quite  sufficient  range  to  be 
obtained  while  the  latter  had  the  great 
advantage  of  allowing  of  the  polar  in- 
stead of  the  linear  form  of  eonstructiou. 
He  wished  to  say  in  conclusion,  that  the 
point  about  which  he  felt  extremely 
gratified  was  that  none  of  the  speakers, 
although  many  of  them  had  had  consid- 
erable exjDerience  concerning  integrators, 
had  taken  any  objection  to  tlie  division 
of  the  subject  he  had  made.  When  he 
began  to  write  the  paper,  his  only  ob- 
ject was  to  endeavor  to  classify  and  put 
in  more  concise  form  the  details  of  the 
subject,  but  he  found  the  utmost  diffi- 
culty in  doing  so.  He  found  out  that 
the  other  classifications  in  use  would  not 
embrace  all  the  instruments,  and  were 
very  unsatisfactory ;  directly,  however, 
the  mode  of  treating  the  whole  question 


201 

which  he  had  employed  in  the  paper,  oc- 
curred to  him,  the  matter  began  to  as- 
sume a  clearer  aspect,  and  he  had  found 
little  trouble  in  grouping  and  explaining 
all  known  integrators  by  its  means  ;  and 
inasmuch  as  the  classification  had  led 
him  to  certain  modifications  of  his  own 
instruments,  he  hoped  that  it  would  not 
be  altogether  without  results  to  other 
workers  on  the  subject. 


202 


Correspondence. 

Mr.  B.  AbdEink-Abakanowiez  submitted 
a  linear  planimeter  of  precision  of  his 
construction  (Fig.  58).  This  instrument 
was  based  on  the  employment  of  a  cylin- 
der C  turning  freely  round  its  axis,  and 
of  a  roller  r,  the  plane  of  which  could  be 
inclined  as  wished  in  respect  to  the  axis 
of  the  cylinder,  and  which  could  be  moved 
along  the  generatrix  of  the  cylinder.  The 
displacement  of  any  point  of  the  cylinder 
was  in  proportion  to  the  tangent  of  the 
angle  that  the  plane  of  the  roller  formed 
with  the  axis  of  the  cylinder.  On  a  bar 
DD,  fixed  on  the  surface  of  a  drawing, 
were  mounted  the  pedestals  S,  S'  carry- 
ing between  their  points  the  cylinder  C. 
The  roller  r  was  mounted  on  a  carrier  B, 
movable  along  the  rod  XX.  Another 
carriage,  A,  supported  a  bar  YY,  perpen- 
dicular to  XX.  The  two  carriers  were 
joined  together  by  the  rod  g,  and  their 
distance  could  be  varied  to  change  the 
value  of  the  constant.     On  the  bar  DD 


203 


204 

as  well  as  on  YY  racks  were  cut,  in  which 
worked  the  pinions^;  and  p' .  The  rod  I, 
which  made  the  plane  of  the  roller  devi- 
ate, was  placed  on  the  plane  of  r  by 
means  of  regulating  screws  mounted  on 
the  arm  R.  In  turning  the  two  pinions 
p  and  p',  it  was  easy  to  follow  exactly 
with  the  pointer  P,  the  curve  bb  of  which 
it  was  desired  to  find  the  superficies. 
There  was  then  imparted  to  the  triangle 
ABP  a  movement  of  translation,  and  at 
the  same  time  its  height  AP  was  made  to 
vary  according  to  the  ordinates  of  the 
given  curve.  The  roller  was  alw^ays 
pressed  by  a  spring  against  the  surface 
of  the  cylinder.  The  number  of  turns  of 
the  cylinder  was  read  on  a  counter  placed 
on  an  endless  screw  v  (this  counter  was 
not  shown  on  Fig.  68),  and  the  fractions 
on  the  drum  T  supplied  with  a  vernier, 
V.  The  integrators  of  Mr.  C.  V.  Boys 
were  founded  on  the  same  principle,  and 
he  took  the  opportunity  of  stating  that 
Mr.  Boj'S  had  found  this  principle  of  in- 
tegration independently  of  him,  and  that 
Mr.  Boys  had  recognized  the  priority  of 


205 

his  invention.  In  other  respects  the 
form  of  the  apparatus  of  Mr.  Boys  differed 
from  that  he  had  constructed. 

The  other  apparatus  he  had  made  on 
the  same  principle,  for  a  special  object 
he  was  now  engaged  in  pursuing.  He 
sought  to  make  apparatus  capable  of 
tracing  the  integral  curve,  and  it  was  to 
this  special  end  he  had  directed  his 
labors.  This  was  briefly  the  problem  : 
given  the  curve  OLD  (Fig.  59),  of  which 
the  equation,  was  y=f  [x),  to  trace  me- 
chanically another  curve  EGF,  of  which 
the  equation  should  heY=J\f  {x)  dx  +  C. 
The  constant=AE.  Since  1878  he  had 
constructed  machines  to  solve  this  prob- 
lem, and  he  had  brought  them  forward 
on  various  occasions.  It  would  occupy 
too  long  a  time  now  to  explain  these  in- 
struments. They  were  made  with  the 
object  of  rendering  service,  particularly 
in  the  art  of  engineering,  and  in  limits 
much  more  extensive  than  could  be  ac- 
complished by  planimeters.  In  reality 
the  integral  curve  could  be  met  with  in 
almost  all   problems   of   statics.     Those 


206 

who  were  familiar  with  the  elegant  pro- 
cesses of  graphic  statics  knew  how  im- 
portant to  the  calculation  of  bridges, 
arches,  statical  moments,  and  of  inertia, 
was  the  outline  traces  by  funicular 
curves.     The  machines  that  he  had  called 


briefly  "  Integraphs  ''  traced  these  curves 
mechanically,  and  to  obtain  them  it  was 
sufficient  to  have  the  same  data  as  for  the 
ordinary  methods  of  graphic  statics.  It 
would  be  possible  to  arrive  at  the  same 
result  with  linear  planimeters,  liut  it 
would  be  necessary  to  operate  by  suc- 
cessive additions,  whilst  the  machines 
would  register  at  each  moment  the  incre- 
ment of  the  integral,  tracing  a  curve  on 


207 


wbich  could  be  made  all  the  necessary 
geometric  operations. 

Major- General  H.  P.  Babbage  remarked 
that  what  most  interested  him  was  the 
contrast  between  arithmetical  calculating- 
machines  and  these  integrators.  In  the 
first  there  was  absolute  accuracy  of  re- 
sult, and  the  same  with  all  operators  ; 
and  there  were  mechanical  means  for  cor- 
recting, to  a  certain  extent,  slackness  of 
the  machinery.  Friction  too  had  to  be 
avoided.  In  the  other  instruments  nearly 
all  this  was  reversed,  and  it  would  seem 
that  with  the  multiplication  of  reliable 
calculating  machines,  all  except  the 
simplest  planimeters  would  become  ob- 
solete. 

Professor  A.  G.  Greenhill  said,  suppose 
O  to  be  the  fixed  center,  GAP  the  planim- 
eter,  A  the  joint  containing  the  sphere 
(Fig.  60). 

1.  Fix  the  joint  A,  and  move  P  to  P, 
on  the  arc  of  a  circle,  center  O ;  then  if 
OA  turned  through  an  angle  0,  the  dial 
at  A  would  register  an  angle  M  d  cos  q), 
if  the  angle  between   AP  and  OA  pro- 


208 

duced    was    q)^    and    M    was    some    cod- 
stant. 

2.  Fix  the  joint  O,  and  move  P,  to  P, 


on  the  arc  of  a  circle,  center  A, ;  the  dial 
at  A  would  not  move. 

3.  Fix  the  joint  A,  and  move  A,  back 
to  A,  and  P,^  to  P3 ;  then  the  dial  at  A 
would  move  back  an  angle  M  d  cos  cp\ 
if  (p'  was  the  angle  between  AP,  and  OA 
produced . 


209 

4.  Fix  the  joint  O,  and  move  P3  to  P 
on  the  arc  of  a  circle,  center  A  ;  the  dial 
would  not  move. 

In  completing  the  circle  of  the  finite 
area  PPjP^P^,  the  dial  would  then  have 
registered  an  angle  Md  (cos  9  — cos  qj'). 

But  the  area  PP^P^P,^  area  PP,Q,Q 

=i  ff  (OP'-OQ^) 

=i  6  («'  +  2  ab  cos  cp-hb'-^r  — 
2  ab  cos  cp'—b"^) 

=.  a  b  6  (cos  cp  —  coH  qj'), 

if  0A  =  (/.  AP  =  />. 

Then  if  "M  —  ab,  the  dial  would  register 
the  area  PP,P,P3. 

Any  irregular  area  must  be  supposed 
to  be  made  up  of  infinitesimal  elements 
formed  in   the  same  manner    as    PP,P^ 

Mr.  E.  Sang  observed  that  in  the 
"  Transactions  of  the  Koyal  Scottish  So- 
ciety of  Arts  "  there  was  a  description  of  a 
plantometer,  by  Arthur  Beverley,  of  Dun- 
edin.  New  Zealand.  In  it  the  rubbing  wheel 
was  guided  in  a  straight  path.  A  very 
simple  analysis  showed  that,  because  the 


210 

tracer  returns  to  its  first  position,  the  re- 
sult was  true,  whatever  might  be  the 
curve  of  this  guide,  so  that  Beverley's 
and  Amsler-Laffon's  were  two  cases  of  a 
general  law.  They  were  identical  in 
principle ;  it  was  not  likely  that,  in  his 
out-of-the-way  position,  Beverley  had 
known  of  the  other. 

Professor  Shaw  remarked  that  he  had 
already  called  attention  to  the  fact  that 
Mr.  Abdank-Abakanowicz  had  anticipated 
Mr.  Boys  in  the  use  of  the  non-slipping 
principle  for  integrators,  but  he  was  glad 
that  the  former  gentleman  had  alluded 
to  the  tracing  of  an  integral  curve  by  in- 
struments acting  on  this  principle.  This 
was  a  problem  of  considerable  interest, 
and  the  author  migbt  mention  that  he 
had,  in  a  j^aper  before  the  Royal  Society, 
called  attention  to  a  particular  case  of  the 
integral  curve.  In  this  case  if  the  section 
of  any  solid  were  taken  on  the  plane  of 
the  paper,  a  curve,  which  he  had  called 
the  curve  of  areas,  might  be  drawn,  the 
ordinates  at  any  point  of  which  repre- 
sented the  area  of  the  cross-section  of  the 


2J1 

solid  at  that  point  perpendicular  to  the 
plane  of  the  paper.  Such  curves  might 
no  doubt  be  drawn  by  an  adaptation  of 
the  instruments  of  Messrs.  Abdank-Aba- 
kanowicz  and  Boys,  and  there  seemed  to 
be  considerable  scope  for  further  investi- 
gation in  that  direction. 

The  author  was  obliged  to  express  his 
disagreement  with  the  opinion  of  Gen- 
eral Babbage,  that  all  integrators  except 
the  simplest  planimeters  would  become 
obsolete  and  give  place  to  arithmetical 
calculating  machines.  Continuous  and 
discontinuous  calculating  machines,  as 
they  had  respectively  been  called,  had 
entirely  different  kinds  of  operations  to 
perform,  and  there  was  a  wide  field  for 
the  employment  of  both.  All  efforts  to 
employ  mere  combinations  of  trains  of 
wheelwork  for  such  operations  as  were 
required  in  continuous  integrators  had 
hitherto  entirely  failed,  and  the  author 
did  not  see  how  it  was  possible  to  deal 
in  this  way  with  the  continuously  varying 
quantities  which  came  into  the  problem. 
No  doubt  the  mechanical  difficulties  were 


212 


great,  but  that  they  av ere  not  insuperable 
was  proved  by  the  daily  use  of  the  disk,  ■ 
globe,  and  cylinder  of  Professor  James  ^ 
Thomson  in  connection  with  tidal  calcu- 
lations and  meteorological  work,  and,  in- 
deed, this  of  itself  was  sufficient  refuta- 
tion of  General  Babbage's  view. 


€ 


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